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G = C5×C22.57C24order 320 = 26·5

Direct product of C5 and C22.57C24

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

Aliases: C5×C22.57C24, C10.1722+ 1+4, C10.1242- 1+4, C4⋊Q818C10, C22⋊Q820C10, C422C29C10, C42.55(C2×C10), C42.C212C10, C4.4D4.9C10, (C2×C10).383C24, (C2×C20).684C23, (C4×C20).296C22, (D4×C10).225C22, C23.26(C22×C10), C22.57(C23×C10), (Q8×C10).188C22, C22.D4.3C10, C2.24(C5×2+ 1+4), C2.16(C5×2- 1+4), (C22×C20).463C22, (C22×C10).109C23, (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×C422C2)⋊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

C1C22 — C5×C22.57C24
C1C2C22C2×C10C2×C20Q8×C10C5×C4⋊Q8 — C5×C22.57C24
C1C22 — C5×C22.57C24
C1C2×C10 — C5×C22.57C24

Generators and relations for C5×C22.57C24
 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 >

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], C422C2 [×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×C422C2 [×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

Smallest permutation representation of C5×C22.57C24
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 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(136 146)(137 147)(138 148)(139 149)(140 150)(141 151)(142 152)(143 153)(144 154)(145 155)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)(76 81)(77 82)(78 83)(79 84)(80 85)(86 91)(87 92)(88 93)(89 94)(90 95)(96 101)(97 102)(98 103)(99 104)(100 105)(106 111)(107 112)(108 113)(109 114)(110 115)(116 121)(117 122)(118 123)(119 124)(120 125)(126 131)(127 132)(128 133)(129 134)(130 135)(136 141)(137 142)(138 143)(139 144)(140 145)(146 151)(147 152)(148 153)(149 154)(150 155)
(1 106 26 96)(2 107 27 97)(3 108 28 98)(4 109 29 99)(5 110 30 100)(6 81 16 91)(7 82 17 92)(8 83 18 93)(9 84 19 94)(10 85 20 95)(11 86 156 76)(12 87 157 77)(13 88 158 78)(14 89 159 79)(15 90 160 80)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(36 126 46 116)(37 127 47 117)(38 128 48 118)(39 129 49 119)(40 130 50 120)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(56 146 66 136)(57 147 67 137)(58 148 68 138)(59 149 69 139)(60 150 70 140)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)
(1 66 26 56)(2 67 27 57)(3 68 28 58)(4 69 29 59)(5 70 30 60)(6 131 16 121)(7 132 17 122)(8 133 18 123)(9 134 19 124)(10 135 20 125)(11 116 156 126)(12 117 157 127)(13 118 158 128)(14 119 159 129)(15 120 160 130)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 91 51 81)(42 92 52 82)(43 93 53 83)(44 94 54 84)(45 95 55 85)(96 136 106 146)(97 137 107 147)(98 138 108 148)(99 139 109 149)(100 140 110 150)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 151 16 141)(7 152 17 142)(8 153 18 143)(9 154 19 144)(10 155 20 145)(11 136 156 146)(12 137 157 147)(13 138 158 148)(14 139 159 149)(15 140 160 150)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(56 81 66 91)(57 82 67 92)(58 83 68 93)(59 84 69 94)(60 85 70 95)(61 76 71 86)(62 77 72 87)(63 78 73 88)(64 79 74 89)(65 80 75 90)(96 131 106 121)(97 132 107 122)(98 133 108 123)(99 134 109 124)(100 135 110 125)(101 126 111 116)(102 127 112 117)(103 128 113 118)(104 129 114 119)(105 130 115 120)
(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(96 111)(97 112)(98 113)(99 114)(100 115)(101 106)(102 107)(103 108)(104 109)(105 110)(116 131)(117 132)(118 133)(119 134)(120 135)(121 126)(122 127)(123 128)(124 129)(125 130)(136 141)(137 142)(138 143)(139 144)(140 145)(146 151)(147 152)(148 153)(149 154)(150 155)

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

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

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

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+42- 1+4C5×2+ 1+4C5×2- 1+4
kernelC5×C22.57C24C5×C22⋊Q8C5×C22.D4C5×C4.4D4C5×C42.C2C5×C422C2C5×C4⋊Q8C22.57C24C22⋊Q8C22.D4C4.4D4C42.C2C422C2C4⋊Q8C10C10C2C2
# reps14212424168481681248

Matrix representation of C5×C22.57C24 in 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] >;

C5×C22.57C24 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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