Copied to
clipboard

G = C5×C23.8Q8order 320 = 26·5

Direct product of C5 and C23.8Q8

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

Aliases: C5×C23.8Q8, C2.4(D4×C20), C22⋊C44C20, (C2×C20).454D4, C10.135(C4×D4), C23.8(C5×Q8), (C23×C4).4C10, (C23×C20).7C2, C23.35(C5×D4), C10.88C22≀C2, C23.15(C2×C20), C24.26(C2×C10), C22.34(D4×C10), (C22×C10).20Q8, C22.12(Q8×C10), C2.C427C10, (C22×C10).155D4, C10.83(C22⋊Q8), C23.58(C22×C10), (C23×C10).86C22, C22.34(C22×C20), (C22×C20).493C22, (C22×C10).449C23, C10.86(C22.D4), (C2×C4⋊C4)⋊2C10, (C2×C4)⋊2(C2×C20), C2.7(C10×C4⋊C4), C222(C5×C4⋊C4), (C10×C4⋊C4)⋊29C2, (C2×C10)⋊8(C4⋊C4), (C2×C20)⋊36(C2×C4), C10.85(C2×C4⋊C4), (C2×C4).99(C5×D4), (C5×C22⋊C4)⋊16C4, C2.2(C5×C22⋊Q8), C2.2(C5×C22≀C2), (C2×C10).601(C2×D4), (C2×C22⋊C4).5C10, (C2×C10).104(C2×Q8), C22.19(C5×C4○D4), (C10×C22⋊C4).11C2, (C22×C4).86(C2×C10), (C2×C10).209(C4○D4), C2.2(C5×C22.D4), (C2×C10).322(C22×C4), (C5×C2.C42)⋊23C2, (C22×C10).150(C2×C4), SmallGroup(320,886)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C23.8Q8
C1C2C22C23C22×C10C22×C20C10×C4⋊C4 — C5×C23.8Q8
C1C22 — C5×C23.8Q8
C1C22×C10 — C5×C23.8Q8

Generators and relations for C5×C23.8Q8
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 370 in 234 conjugacy classes, 106 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C24, C20 [×10], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C2×C20 [×8], C2×C20 [×22], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.8Q8, C5×C22⋊C4 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×6], C23×C10, C5×C2.C42 [×2], C10×C22⋊C4 [×2], C10×C4⋊C4 [×2], C23×C20, C5×C23.8Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×6], Q8 [×2], C23, C10 [×7], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C20 [×4], C2×C10 [×7], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×C20 [×6], C5×D4 [×6], C5×Q8 [×2], C22×C10, C23.8Q8, C5×C4⋊C4 [×4], C22×C20, D4×C10 [×3], Q8×C10, C5×C4○D4 [×2], C10×C4⋊C4, D4×C20 [×2], C5×C22≀C2, C5×C22⋊Q8 [×2], C5×C22.D4, C5×C23.8Q8

Smallest permutation representation of C5×C23.8Q8
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 54)(2 55)(3 51)(4 52)(5 53)(6 31)(7 32)(8 33)(9 34)(10 35)(11 44)(12 45)(13 41)(14 42)(15 43)(16 25)(17 21)(18 22)(19 23)(20 24)(26 40)(27 36)(28 37)(29 38)(30 39)(46 70)(47 66)(48 67)(49 68)(50 69)(56 65)(57 61)(58 62)(59 63)(60 64)(71 80)(72 76)(73 77)(74 78)(75 79)(81 94)(82 95)(83 91)(84 92)(85 93)(86 110)(87 106)(88 107)(89 108)(90 109)(96 117)(97 118)(98 119)(99 120)(100 116)(101 114)(102 115)(103 111)(104 112)(105 113)(121 127)(122 128)(123 129)(124 130)(125 126)(131 148)(132 149)(133 150)(134 146)(135 147)(136 157)(137 158)(138 159)(139 160)(140 156)(141 154)(142 155)(143 151)(144 152)(145 153)
(1 54)(2 55)(3 51)(4 52)(5 53)(6 19)(7 20)(8 16)(9 17)(10 18)(11 44)(12 45)(13 41)(14 42)(15 43)(21 34)(22 35)(23 31)(24 32)(25 33)(26 40)(27 36)(28 37)(29 38)(30 39)(46 70)(47 66)(48 67)(49 68)(50 69)(56 65)(57 61)(58 62)(59 63)(60 64)(71 80)(72 76)(73 77)(74 78)(75 79)(81 94)(82 95)(83 91)(84 92)(85 93)(86 110)(87 106)(88 107)(89 108)(90 109)(96 105)(97 101)(98 102)(99 103)(100 104)(111 120)(112 116)(113 117)(114 118)(115 119)(121 134)(122 135)(123 131)(124 132)(125 133)(126 150)(127 146)(128 147)(129 148)(130 149)(136 145)(137 141)(138 142)(139 143)(140 144)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 23)(7 24)(8 25)(9 21)(10 22)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 34)(18 35)(19 31)(20 32)(26 43)(27 44)(28 45)(29 41)(30 42)(46 53)(47 54)(48 55)(49 51)(50 52)(56 73)(57 74)(58 75)(59 71)(60 72)(61 78)(62 79)(63 80)(64 76)(65 77)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 113)(97 114)(98 115)(99 111)(100 112)(101 118)(102 119)(103 120)(104 116)(105 117)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 153)(137 154)(138 155)(139 151)(140 152)(141 158)(142 159)(143 160)(144 156)(145 157)
(1 65 12 81)(2 61 13 82)(3 62 14 83)(4 63 15 84)(5 64 11 85)(6 100 144 133)(7 96 145 134)(8 97 141 135)(9 98 142 131)(10 99 143 132)(16 101 137 122)(17 102 138 123)(18 103 139 124)(19 104 140 125)(20 105 136 121)(21 115 159 129)(22 111 160 130)(23 112 156 126)(24 113 157 127)(25 114 158 128)(26 90 50 71)(27 86 46 72)(28 87 47 73)(29 88 48 74)(30 89 49 75)(31 116 152 150)(32 117 153 146)(33 118 154 147)(34 119 155 148)(35 120 151 149)(36 110 70 76)(37 106 66 77)(38 107 67 78)(39 108 68 79)(40 109 69 80)(41 95 55 57)(42 91 51 58)(43 92 52 59)(44 93 53 60)(45 94 54 56)
(1 105 12 121)(2 101 13 122)(3 102 14 123)(4 103 15 124)(5 104 11 125)(6 85 144 64)(7 81 145 65)(8 82 141 61)(9 83 142 62)(10 84 143 63)(16 95 137 57)(17 91 138 58)(18 92 139 59)(19 93 140 60)(20 94 136 56)(21 108 159 79)(22 109 160 80)(23 110 156 76)(24 106 157 77)(25 107 158 78)(26 130 50 111)(27 126 46 112)(28 127 47 113)(29 128 48 114)(30 129 49 115)(31 86 152 72)(32 87 153 73)(33 88 154 74)(34 89 155 75)(35 90 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(41 135 55 97)(42 131 51 98)(43 132 52 99)(44 133 53 100)(45 134 54 96)

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

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

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

140 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B5C5D10A···10AB10AC···10AR20A···20AF20AG···20BL
order12···222224···44···4555510···1010···1020···2020···20
size11···122222···24···411111···12···22···24···4

140 irreducible representations

dim11111111111122222222
type+++++++-
imageC1C2C2C2C2C4C5C10C10C10C10C20D4D4Q8C4○D4C5×D4C5×D4C5×Q8C5×C4○D4
kernelC5×C23.8Q8C5×C2.C42C10×C22⋊C4C10×C4⋊C4C23×C20C5×C22⋊C4C23.8Q8C2.C42C2×C22⋊C4C2×C4⋊C4C23×C4C22⋊C4C2×C20C22×C10C22×C10C2×C10C2×C4C23C23C22
# reps12221848884324224168816

Matrix representation of C5×C23.8Q8 in GL6(𝔽41)

100000
010000
001000
000100
0000160
0000016
,
4000000
0400000
001000
000100
0000400
000041
,
4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
001000
000100
0000400
0000040
,
1390000
0400000
000100
0040000
0000320
0000032
,
17120000
17240000
00151500
00152600
00003739
0000294

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,4,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[17,17,0,0,0,0,12,24,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,37,29,0,0,0,0,39,4] >;

C5×C23.8Q8 in GAP, Magma, Sage, TeX

C_5\times C_2^3._8Q_8
% in TeX

G:=Group("C5xC2^3.8Q8");
// GroupNames label

G:=SmallGroup(320,886);
// by ID

G=gap.SmallGroup(320,886);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1128,1766]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations

׿
×
𝔽