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G = C5×C22.57C24  order 320 = 26·5

Direct product of C5 and C22.57C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C22.57C24, C10.1242- (1+4), C10.1722+ (1+4), C4⋊Q8⋊18C10, C22⋊Q8⋊20C10, C42⋊2C2⋊9C10, C42.55(C2×C10), C42.C2⋊12C10, C4.4D4.9C10, (C2×C10).383C24, (C4×C20).296C22, (C2×C20).684C23, (D4×C10).225C22, C23.26(C22×C10), C22.57(C23×C10), (Q8×C10).188C22, C22.D4.3C10, C2.16(C5×2- (1+4)), C2.24(C5×2+ (1+4)), (C22×C10).109C23, (C22×C20).463C22, (C5×C4⋊Q8)⋊39C2, C4⋊C4.35(C2×C10), (C5×C22⋊Q8)⋊47C2, (C2×D4).38(C2×C10), C22⋊C4.8(C2×C10), (C2×Q8).31(C2×C10), (C5×C42.C2)⋊29C2, (C5×C42⋊2C2)⋊20C2, (C5×C4⋊C4).252C22, (C2×C4).43(C22×C10), (C22×C4).74(C2×C10), (C5×C4.4D4).18C2, (C5×C22⋊C4).93C22, (C5×C22.D4).6C2, SmallGroup(320,1565)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.57C24
Chief series
C1 — C2 — C22 — C2×C10 — C2×C20 — Q8×C10 — C5×C4⋊Q8 — C5×C22.57C24
Lower central
C1 — C22 — C5×C22.57C24
Upper central
C1 — C2×C10 — C5×C22.57C24

Subgroups: 282 in 196 conjugacy classes, 142 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4, C2×C4 [×12], C2×C4 [×2], D4, Q8 [×3], C23 [×2], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C20 [×13], C2×C10, C2×C10 [×6], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], C2×C20, C2×C20 [×12], C2×C20 [×2], C5×D4, C5×Q8 [×3], C22×C10 [×2], C22.57C24, C4×C20, C4×C20 [×2], C5×C22⋊C4 [×10], C5×C4⋊C4 [×16], C22×C20 [×2], D4×C10, Q8×C10, Q8×C10 [×2], C5×C22⋊Q8 [×4], C5×C22.D4 [×2], C5×C4.4D4, C5×C42.C2 [×2], C5×C42⋊2C2 [×4], C5×C4⋊Q8 [×2], C5×C22.57C24

Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], 2+ (1+4), 2- (1+4) [×2], C22×C10 [×15], C22.57C24, C23×C10, C5×2+ (1+4), C5×2- (1+4) [×2], C5×C22.57C24

Generators and relations
 G = < a,b,c,d,e,f,g | a5=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Smallest permutation representation
►On 160 points
Generators in S160
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)

(1 35)(2 31)(3 32)(4 33)(5 34)(6 160)(7 156)(8 157)(9 158)(10 159)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 115)(97 111)(98 112)(99 113)(100 114)(101 107)(102 108)(103 109)(104 110)(105 106)(116 135)(117 131)(118 132)(119 133)(120 134)(121 127)(122 128)(123 129)(124 130)(125 126)(136 155)(137 151)(138 152)(139 153)(140 154)(141 147)(142 148)(143 149)(144 150)(145 146)

(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 157)(12 158)(13 159)(14 160)(15 156)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)(76 81)(77 82)(78 83)(79 84)(80 85)(86 94)(87 95)(88 91)(89 92)(90 93)(96 101)(97 102)(98 103)(99 104)(100 105)(106 114)(107 115)(108 111)(109 112)(110 113)(116 121)(117 122)(118 123)(119 124)(120 125)(126 134)(127 135)(128 131)(129 132)(130 133)(136 141)(137 142)(138 143)(139 144)(140 145)(146 154)(147 155)(148 151)(149 152)(150 153)

(1 115 35 96)(2 111 31 97)(3 112 32 98)(4 113 33 99)(5 114 34 100)(6 94 160 80)(7 95 156 76)(8 91 157 77)(9 92 158 78)(10 93 159 79)(11 82 17 88)(12 83 18 89)(13 84 19 90)(14 85 20 86)(15 81 16 87)(21 107 27 101)(22 108 28 102)(23 109 29 103)(24 110 30 104)(25 106 26 105)(36 135 55 116)(37 131 51 117)(38 132 52 118)(39 133 53 119)(40 134 54 120)(41 127 47 121)(42 128 48 122)(43 129 49 123)(44 130 50 124)(45 126 46 125)(56 155 75 136)(57 151 71 137)(58 152 72 138)(59 153 73 139)(60 154 74 140)(61 147 67 141)(62 148 68 142)(63 149 69 143)(64 150 70 144)(65 146 66 145)

(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 120 160 134)(7 116 156 135)(8 117 157 131)(9 118 158 132)(10 119 159 133)(11 128 17 122)(12 129 18 123)(13 130 19 124)(14 126 20 125)(15 127 16 121)(21 67 27 61)(22 68 28 62)(23 69 29 63)(24 70 30 64)(25 66 26 65)(36 95 55 76)(37 91 51 77)(38 92 52 78)(39 93 53 79)(40 94 54 80)(41 87 47 81)(42 88 48 82)(43 89 49 83)(44 90 50 84)(45 86 46 85)(96 136 115 155)(97 137 111 151)(98 138 112 152)(99 139 113 153)(100 140 114 154)(101 141 107 147)(102 142 108 148)(103 143 109 149)(104 144 110 150)(105 145 106 146)

(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 140 160 154)(7 136 156 155)(8 137 157 151)(9 138 158 152)(10 139 159 153)(11 148 17 142)(12 149 18 143)(13 150 19 144)(14 146 20 145)(15 147 16 141)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)(56 81 75 87)(57 82 71 88)(58 83 72 89)(59 84 73 90)(60 85 74 86)(61 76 67 95)(62 77 68 91)(63 78 69 92)(64 79 70 93)(65 80 66 94)(96 127 115 121)(97 128 111 122)(98 129 112 123)(99 130 113 124)(100 126 114 125)(101 135 107 116)(102 131 108 117)(103 132 109 118)(104 133 110 119)(105 134 106 120)

(6 20)(7 16)(8 17)(9 18)(10 19)(11 157)(12 158)(13 159)(14 160)(15 156)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 107)(97 108)(98 109)(99 110)(100 106)(101 115)(102 111)(103 112)(104 113)(105 114)(116 127)(117 128)(118 129)(119 130)(120 126)(121 135)(122 131)(123 132)(124 133)(125 134)(136 141)(137 142)(138 143)(139 144)(140 145)(146 154)(147 155)(148 151)(149 152)(150 153)
 
G:=sub<Sym(160)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,115,35,96)(2,111,31,97)(3,112,32,98)(4,113,33,99)(5,114,34,100)(6,94,160,80)(7,95,156,76)(8,91,157,77)(9,92,158,78)(10,93,159,79)(11,82,17,88)(12,83,18,89)(13,84,19,90)(14,85,20,86)(15,81,16,87)(21,107,27,101)(22,108,28,102)(23,109,29,103)(24,110,30,104)(25,106,26,105)(36,135,55,116)(37,131,51,117)(38,132,52,118)(39,133,53,119)(40,134,54,120)(41,127,47,121)(42,128,48,122)(43,129,49,123)(44,130,50,124)(45,126,46,125)(56,155,75,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,147,67,141)(62,148,68,142)(63,149,69,143)(64,150,70,144)(65,146,66,145), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,120,160,134)(7,116,156,135)(8,117,157,131)(9,118,158,132)(10,119,159,133)(11,128,17,122)(12,129,18,123)(13,130,19,124)(14,126,20,125)(15,127,16,121)(21,67,27,61)(22,68,28,62)(23,69,29,63)(24,70,30,64)(25,66,26,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,87,47,81)(42,88,48,82)(43,89,49,83)(44,90,50,84)(45,86,46,85)(96,136,115,155)(97,137,111,151)(98,138,112,152)(99,139,113,153)(100,140,114,154)(101,141,107,147)(102,142,108,148)(103,143,109,149)(104,144,110,150)(105,145,106,146), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,81,75,87)(57,82,71,88)(58,83,72,89)(59,84,73,90)(60,85,74,86)(61,76,67,95)(62,77,68,91)(63,78,69,92)(64,79,70,93)(65,80,66,94)(96,127,115,121)(97,128,111,122)(98,129,112,123)(99,130,113,124)(100,126,114,125)(101,135,107,116)(102,131,108,117)(103,132,109,118)(104,133,110,119)(105,134,106,120), (6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,107)(97,108)(98,109)(99,110)(100,106)(101,115)(102,111)(103,112)(104,113)(105,114)(116,127)(117,128)(118,129)(119,130)(120,126)(121,135)(122,131)(123,132)(124,133)(125,134)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,115,35,96)(2,111,31,97)(3,112,32,98)(4,113,33,99)(5,114,34,100)(6,94,160,80)(7,95,156,76)(8,91,157,77)(9,92,158,78)(10,93,159,79)(11,82,17,88)(12,83,18,89)(13,84,19,90)(14,85,20,86)(15,81,16,87)(21,107,27,101)(22,108,28,102)(23,109,29,103)(24,110,30,104)(25,106,26,105)(36,135,55,116)(37,131,51,117)(38,132,52,118)(39,133,53,119)(40,134,54,120)(41,127,47,121)(42,128,48,122)(43,129,49,123)(44,130,50,124)(45,126,46,125)(56,155,75,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,147,67,141)(62,148,68,142)(63,149,69,143)(64,150,70,144)(65,146,66,145), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,120,160,134)(7,116,156,135)(8,117,157,131)(9,118,158,132)(10,119,159,133)(11,128,17,122)(12,129,18,123)(13,130,19,124)(14,126,20,125)(15,127,16,121)(21,67,27,61)(22,68,28,62)(23,69,29,63)(24,70,30,64)(25,66,26,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,87,47,81)(42,88,48,82)(43,89,49,83)(44,90,50,84)(45,86,46,85)(96,136,115,155)(97,137,111,151)(98,138,112,152)(99,139,113,153)(100,140,114,154)(101,141,107,147)(102,142,108,148)(103,143,109,149)(104,144,110,150)(105,145,106,146), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,81,75,87)(57,82,71,88)(58,83,72,89)(59,84,73,90)(60,85,74,86)(61,76,67,95)(62,77,68,91)(63,78,69,92)(64,79,70,93)(65,80,66,94)(96,127,115,121)(97,128,111,122)(98,129,112,123)(99,130,113,124)(100,126,114,125)(101,135,107,116)(102,131,108,117)(103,132,109,118)(104,133,110,119)(105,134,106,120), (6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,107)(97,108)(98,109)(99,110)(100,106)(101,115)(102,111)(103,112)(104,113)(105,114)(116,127)(117,128)(118,129)(119,130)(120,126)(121,135)(122,131)(123,132)(124,133)(125,134)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153) );
 
G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,160),(7,156),(8,157),(9,158),(10,159),(11,17),(12,18),(13,19),(14,20),(15,16),(21,27),(22,28),(23,29),(24,30),(25,26),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66),(76,95),(77,91),(78,92),(79,93),(80,94),(81,87),(82,88),(83,89),(84,90),(85,86),(96,115),(97,111),(98,112),(99,113),(100,114),(101,107),(102,108),(103,109),(104,110),(105,106),(116,135),(117,131),(118,132),(119,133),(120,134),(121,127),(122,128),(123,129),(124,130),(125,126),(136,155),(137,151),(138,152),(139,153),(140,154),(141,147),(142,148),(143,149),(144,150),(145,146)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,157),(12,158),(13,159),(14,160),(15,156),(26,34),(27,35),(28,31),(29,32),(30,33),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53),(56,61),(57,62),(58,63),(59,64),(60,65),(66,74),(67,75),(68,71),(69,72),(70,73),(76,81),(77,82),(78,83),(79,84),(80,85),(86,94),(87,95),(88,91),(89,92),(90,93),(96,101),(97,102),(98,103),(99,104),(100,105),(106,114),(107,115),(108,111),(109,112),(110,113),(116,121),(117,122),(118,123),(119,124),(120,125),(126,134),(127,135),(128,131),(129,132),(130,133),(136,141),(137,142),(138,143),(139,144),(140,145),(146,154),(147,155),(148,151),(149,152),(150,153)], [(1,115,35,96),(2,111,31,97),(3,112,32,98),(4,113,33,99),(5,114,34,100),(6,94,160,80),(7,95,156,76),(8,91,157,77),(9,92,158,78),(10,93,159,79),(11,82,17,88),(12,83,18,89),(13,84,19,90),(14,85,20,86),(15,81,16,87),(21,107,27,101),(22,108,28,102),(23,109,29,103),(24,110,30,104),(25,106,26,105),(36,135,55,116),(37,131,51,117),(38,132,52,118),(39,133,53,119),(40,134,54,120),(41,127,47,121),(42,128,48,122),(43,129,49,123),(44,130,50,124),(45,126,46,125),(56,155,75,136),(57,151,71,137),(58,152,72,138),(59,153,73,139),(60,154,74,140),(61,147,67,141),(62,148,68,142),(63,149,69,143),(64,150,70,144),(65,146,66,145)], [(1,75,35,56),(2,71,31,57),(3,72,32,58),(4,73,33,59),(5,74,34,60),(6,120,160,134),(7,116,156,135),(8,117,157,131),(9,118,158,132),(10,119,159,133),(11,128,17,122),(12,129,18,123),(13,130,19,124),(14,126,20,125),(15,127,16,121),(21,67,27,61),(22,68,28,62),(23,69,29,63),(24,70,30,64),(25,66,26,65),(36,95,55,76),(37,91,51,77),(38,92,52,78),(39,93,53,79),(40,94,54,80),(41,87,47,81),(42,88,48,82),(43,89,49,83),(44,90,50,84),(45,86,46,85),(96,136,115,155),(97,137,111,151),(98,138,112,152),(99,139,113,153),(100,140,114,154),(101,141,107,147),(102,142,108,148),(103,143,109,149),(104,144,110,150),(105,145,106,146)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,140,160,154),(7,136,156,155),(8,137,157,151),(9,138,158,152),(10,139,159,153),(11,148,17,142),(12,149,18,143),(13,150,19,144),(14,146,20,145),(15,147,16,141),(21,47,27,41),(22,48,28,42),(23,49,29,43),(24,50,30,44),(25,46,26,45),(56,81,75,87),(57,82,71,88),(58,83,72,89),(59,84,73,90),(60,85,74,86),(61,76,67,95),(62,77,68,91),(63,78,69,92),(64,79,70,93),(65,80,66,94),(96,127,115,121),(97,128,111,122),(98,129,112,123),(99,130,113,124),(100,126,114,125),(101,135,107,116),(102,131,108,117),(103,132,109,118),(104,133,110,119),(105,134,106,120)], [(6,20),(7,16),(8,17),(9,18),(10,19),(11,157),(12,158),(13,159),(14,160),(15,156),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66),(76,95),(77,91),(78,92),(79,93),(80,94),(81,87),(82,88),(83,89),(84,90),(85,86),(96,107),(97,108),(98,109),(99,110),(100,106),(101,115),(102,111),(103,112),(104,113),(105,114),(116,127),(117,128),(118,129),(119,130),(120,126),(121,135),(122,131),(123,132),(124,133),(125,134),(136,141),(137,142),(138,143),(139,144),(140,145),(146,154),(147,155),(148,151),(149,152),(150,153)])
 

Matrix representation ►G ⊆ GL8(𝔽41)

10000000
01000000
00100000
00010000
000037000
000003700
000000370
000000037
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
103900000
004010000
004000000
014000000
000032000
00000900
00008090
000008032
,
3830000000
383000000
01512260000
381515290000
0000317390
00003410039
00002601034
0000026731
,
139000000
040000000
1400400000
1404000000
00000100
000040000
000003101
0000310400
,
10000000
01000000
104000000
100400000
00001000
00000100
0000317400
00003410040

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,39,40,40,40,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,32,0,8,0,0,0,0,0,0,9,0,8,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32],[38,38,0,38,0,0,0,0,30,3,15,15,0,0,0,0,0,0,12,15,0,0,0,0,0,0,26,29,0,0,0,0,0,0,0,0,31,34,26,0,0,0,0,0,7,10,0,26,0,0,0,0,39,0,10,7,0,0,0,0,0,39,34,31],[1,0,1,1,0,0,0,0,39,40,40,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,31,0,0,0,0,1,0,31,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,31,34,0,0,0,0,0,1,7,10,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
 

95 conjugacy classes

class 1 2A2B2C2D2E4A···4M5A5B5C5D10A···10L10M···10T20A···20AZ
order1222224···4555510···1010···1020···20
size1111444···411111···14···44···4

95 irreducible representations

dim111111111111114444
type++++++++-
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C102+ (1+4)2- (1+4)C5×2+ (1+4)C5×2- (1+4)
kernelC5×C22.57C24C5×C22⋊Q8C5×C22.D4C5×C4.4D4C5×C42.C2C5×C42⋊2C2C5×C4⋊Q8C22.57C24C22⋊Q8C22.D4C4.4D4C42.C2C42⋊2C2C4⋊Q8C10C10C2C2
# reps14212424168481681248

In GAP, Magma, Sage, TeX 

C_5\times C_2^2._{57}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.57C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1565);
 
// by ID
 
G=gap.SmallGroup(320,1565);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446,2571,436,6947,1242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=g^2=1,d^2=e^2=f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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