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G = C5×C23.32C23  order 320 = 26·5

Direct product of C5 and C23.32C23

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

Aliases: C5×C23.32C23, C10.1092- (1+4), (C2×Q8)⋊9C20, (C4×Q8)⋊5C10, (Q8×C20)⋊25C2, (Q8×C10)⋊29C4, Q8.8(C2×C20), C2.8(C23×C20), C42.32(C2×C10), C4.20(C22×C20), C10.81(C23×C4), (C22×Q8).7C10, C20.224(C22×C4), (C2×C20).710C23, (C2×C10).339C24, (C4×C20).275C22, C42⋊C2.10C10, C2.1(C5×2- (1+4)), C23.31(C22×C10), C22.12(C23×C10), C22.11(C22×C20), (Q8×C10).284C22, (C22×C10).255C23, (C22×C20).442C22, (Q8×C2×C10).17C2, C4⋊C4.82(C2×C10), (C2×C4).31(C2×C20), (C5×Q8).47(C2×C4), (C2×C20).376(C2×C4), (C2×Q8).72(C2×C10), C22⋊C4.29(C2×C10), (C5×C4⋊C4).407C22, (C22×C4).52(C2×C10), (C2×C10).266(C22×C4), (C2×C4).135(C22×C10), (C5×C42⋊C2).24C2, (C5×C22⋊C4).160C22, SmallGroup(320,1521)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C5×C23.32C23
Chief series
C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4 — Q8×C20 — C5×C23.32C23
Lower central
C1 — C2 — C5×C23.32C23
Upper central
C1 — C2×C10 — C5×C23.32C23

Subgroups: 290 in 266 conjugacy classes, 242 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×2], C5, C2×C4 [×26], Q8 [×16], C23, C10, C10 [×2], C10 [×2], C42 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×12], C20 [×12], C20 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2 [×6], C4×Q8 [×8], C22×Q8, C2×C20 [×26], C5×Q8 [×16], C22×C10, C23.32C23, C4×C20 [×12], C5×C22⋊C4 [×4], C5×C4⋊C4 [×12], C22×C20 [×3], Q8×C10 [×12], C5×C42⋊C2 [×6], Q8×C20 [×8], Q8×C2×C10, C5×C23.32C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C23×C4, 2- (1+4) [×2], C2×C20 [×28], C22×C10 [×15], C23.32C23, C22×C20 [×14], C23×C10, C23×C20, C5×2- (1+4) [×2], C5×C23.32C23

Generators and relations
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=1, e2=d, f2=g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

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 21)(2 22)(3 23)(4 24)(5 25)(6 14)(7 15)(8 11)(9 12)(10 13)(16 156)(17 157)(18 158)(19 159)(20 160)(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)(96 107)(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)(136 147)(137 148)(138 149)(139 150)(140 146)(141 155)(142 151)(143 152)(144 153)(145 154)

(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)

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

100000
0370000
001000
000100
000010
000001
,
100000
0400000
0040000
0004000
000010
0036501
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
010000
0040000
0004000
0000400
0000040
,
900000
010000
000010
00536139
0040000
00281305
,
100000
0400000
00261500
00151500
00734030
0070150
,
4000000
0400000
000100
0040000
00536139
0050140

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,37,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,36,0,0,0,40,0,5,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,40,28,0,0,0,36,0,13,0,0,1,1,0,0,0,0,0,39,0,5],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,26,15,7,7,0,0,15,15,34,0,0,0,0,0,0,15,0,0,0,0,30,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,5,5,0,0,1,0,36,0,0,0,0,0,1,1,0,0,0,0,39,40] >;
 

170 conjugacy classes

class 1 2A2B2C2D2E4A···4AB5A5B5C5D10A···10L10M···10T20A···20DH
order1222224···4555510···1010···1020···20
size1111222···211111···12···22···2

170 irreducible representations

dim111111111144
type++++-
imageC1C2C2C2C4C5C10C10C10C202- (1+4)C5×2- (1+4)
kernelC5×C23.32C23C5×C42⋊C2Q8×C20Q8×C2×C10Q8×C10C23.32C23C42⋊C2C4×Q8C22×Q8C2×Q8C10C2
# reps1681164243246428

In GAP, Magma, Sage, TeX 

C_5\times C_2^3._{32}C_2^3
 
% in TeX
 
G:=Group("C5xC2^3.32C2^3");
 
// GroupNames label
 
G:=SmallGroup(320,1521);
 
// by ID
 
G=gap.SmallGroup(320,1521);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,568,891,436,2467]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=1,e^2=d,f^2=g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

׿
×
⋊
ℤ
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