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G = C5×C23.38D4order 320 = 26·5

Direct product of C5 and C23.38D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C23.38D4, (C2×Q8)⋊6C20, (Q8×C10)⋊26C4, C4.56(D4×C10), Q8.6(C2×C20), C20.463(C2×D4), (C2×C20).316D4, C4.6(C22×C20), C23.38(C5×D4), Q8⋊C415C10, C22.46(D4×C10), (C22×Q8).5C10, (C2×C40).323C22, (C2×C20).895C23, C20.210(C22×C4), C42⋊C2.4C10, (C22×C10).160D4, C20.130(C22⋊C4), (C10×M4(2)).30C2, (C2×M4(2)).12C10, (Q8×C10).253C22, C10.127(C8.C22), (C22×C20).412C22, (C2×C4).24(C5×D4), (Q8×C2×C10).15C2, C4⋊C4.40(C2×C10), (C2×C8).48(C2×C10), (C2×C4).22(C2×C20), C4.15(C5×C22⋊C4), (C5×Q8).45(C2×C4), C2.2(C5×C8.C22), (C2×C20).368(C2×C4), (C5×Q8⋊C4)⋊38C2, (C2×C10).622(C2×D4), C2.22(C10×C22⋊C4), (C2×Q8).38(C2×C10), (C5×C4⋊C4).361C22, C10.151(C2×C22⋊C4), (C22×C4).31(C2×C10), (C2×C4).70(C22×C10), C22.21(C5×C22⋊C4), (C5×C42⋊C2).18C2, (C2×C10).146(C22⋊C4), SmallGroup(320,920)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C23.38D4
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×Q8⋊C4 — C5×C23.38D4
C1C2C4 — C5×C23.38D4
C1C2×C10C22×C20 — C5×C23.38D4

Generators and relations for C5×C23.38D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 226 in 150 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×6], C23, C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×6], C2×Q8 [×3], C20 [×2], C20 [×2], C20 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×8], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C23.38D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C2×C40 [×2], C5×M4(2) [×2], C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], C5×Q8⋊C4 [×4], C5×C42⋊C2, C10×M4(2), Q8×C2×C10, C5×C23.38D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C8.C22 [×2], C2×C20 [×6], C5×D4 [×4], C22×C10, C23.38D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C10×C22⋊C4, C5×C8.C22 [×2], C5×C23.38D4

Smallest permutation representation of C5×C23.38D4
On 160 points
Generators in S160
(1 127 159 39 151)(2 128 160 40 152)(3 121 153 33 145)(4 122 154 34 146)(5 123 155 35 147)(6 124 156 36 148)(7 125 157 37 149)(8 126 158 38 150)(9 113 137 17 129)(10 114 138 18 130)(11 115 139 19 131)(12 116 140 20 132)(13 117 141 21 133)(14 118 142 22 134)(15 119 143 23 135)(16 120 144 24 136)(25 89 56 81 44)(26 90 49 82 45)(27 91 50 83 46)(28 92 51 84 47)(29 93 52 85 48)(30 94 53 86 41)(31 95 54 87 42)(32 96 55 88 43)(57 78 111 70 103)(58 79 112 71 104)(59 80 105 72 97)(60 73 106 65 98)(61 74 107 66 99)(62 75 108 67 100)(63 76 109 68 101)(64 77 110 69 102)
(1 85)(2 82)(3 87)(4 84)(5 81)(6 86)(7 83)(8 88)(9 70)(10 67)(11 72)(12 69)(13 66)(14 71)(15 68)(16 65)(17 78)(18 75)(19 80)(20 77)(21 74)(22 79)(23 76)(24 73)(25 155)(26 160)(27 157)(28 154)(29 159)(30 156)(31 153)(32 158)(33 95)(34 92)(35 89)(36 94)(37 91)(38 96)(39 93)(40 90)(41 124)(42 121)(43 126)(44 123)(45 128)(46 125)(47 122)(48 127)(49 152)(50 149)(51 146)(52 151)(53 148)(54 145)(55 150)(56 147)(57 137)(58 142)(59 139)(60 144)(61 141)(62 138)(63 143)(64 140)(97 115)(98 120)(99 117)(100 114)(101 119)(102 116)(103 113)(104 118)(105 131)(106 136)(107 133)(108 130)(109 135)(110 132)(111 129)(112 134)
(1 85)(2 86)(3 87)(4 88)(5 81)(6 82)(7 83)(8 84)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 153)(32 154)(33 95)(34 96)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 128)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 148)(50 149)(51 150)(52 151)(53 152)(54 145)(55 146)(56 147)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 113)(104 114)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 129)(112 130)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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)
(1 104 85 114)(2 117 86 99)(3 102 87 120)(4 115 88 97)(5 100 81 118)(6 113 82 103)(7 98 83 116)(8 119 84 101)(9 49 70 148)(10 151 71 52)(11 55 72 146)(12 149 65 50)(13 53 66 152)(14 147 67 56)(15 51 68 150)(16 145 69 54)(17 26 78 156)(18 159 79 29)(19 32 80 154)(20 157 73 27)(21 30 74 160)(22 155 75 25)(23 28 76 158)(24 153 77 31)(33 110 95 136)(34 131 96 105)(35 108 89 134)(36 129 90 111)(37 106 91 132)(38 135 92 109)(39 112 93 130)(40 133 94 107)(41 61 128 141)(42 144 121 64)(43 59 122 139)(44 142 123 62)(45 57 124 137)(46 140 125 60)(47 63 126 143)(48 138 127 58)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4L5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20P20Q···20AV40A···40P
order12222244444···45555888810···1010···1020···2020···2040···40
size11112222224···4111144441···12···22···24···44···4

110 irreducible representations

dim111111111111222244
type+++++++-
imageC1C2C2C2C2C4C5C10C10C10C10C20D4D4C5×D4C5×D4C8.C22C5×C8.C22
kernelC5×C23.38D4C5×Q8⋊C4C5×C42⋊C2C10×M4(2)Q8×C2×C10Q8×C10C23.38D4Q8⋊C4C42⋊C2C2×M4(2)C22×Q8C2×Q8C2×C20C22×C10C2×C4C23C10C2
# reps141118416444323112428

Matrix representation of C5×C23.38D4 in GL6(𝔽41)

100000
010000
0010000
0001000
0000100
0000010
,
100000
010000
001011
000177
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
9300000
11320000
002114158
002416034
0013342323
0027202222
,
3090000
32110000
001000
00702425
0039401616
00012424

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,7,40,0,0,0,1,7,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[9,11,0,0,0,0,30,32,0,0,0,0,0,0,21,24,13,27,0,0,14,16,34,20,0,0,15,0,23,22,0,0,8,34,23,22],[30,32,0,0,0,0,9,11,0,0,0,0,0,0,1,7,39,0,0,0,0,0,40,1,0,0,0,24,16,24,0,0,0,25,16,24] >;

C5×C23.38D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{38}D_4
% in TeX

G:=Group("C5xC2^3.38D4");
// GroupNames label

G:=SmallGroup(320,920);
// by ID

G=gap.SmallGroup(320,920);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,1731,856,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*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^3>;
// generators/relations

׿
×
𝔽