Copied to
clipboard

G = C5×C23.48D4order 320 = 26·5

Direct product of C5 and C23.48D4

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

Aliases: C5×C23.48D4, C2.D88C10, C2.8(C10×Q16), Q8⋊C47C10, (C2×C20).339D4, C22⋊C8.4C10, (C2×C10).12Q16, C10.55(C2×Q16), C23.48(C5×D4), C22⋊Q8.6C10, C22.3(C5×Q16), C20.321(C4○D4), (C2×C20).940C23, (C2×C40).263C22, (C22×C10).170D4, C22.105(D4×C10), C10.144(C8⋊C22), (Q8×C10).170C22, (C22×C20).432C22, C10.99(C22.D4), (C5×C2.D8)⋊23C2, C4.33(C5×C4○D4), (C2×C4).40(C5×D4), (C10×C4⋊C4).46C2, (C2×C4⋊C4).17C10, C4⋊C4.61(C2×C10), (C2×C8).10(C2×C10), C2.19(C5×C8⋊C22), (C5×Q8⋊C4)⋊30C2, (C2×C10).661(C2×D4), (C5×C22⋊C8).13C2, (C2×Q8).14(C2×C10), (C5×C22⋊Q8).16C2, (C5×C4⋊C4).384C22, (C22×C4).50(C2×C10), (C2×C4).115(C22×C10), C2.15(C5×C22.D4), SmallGroup(320,985)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C23.48D4
C1C2C4C2×C4C2×C20Q8×C10C5×C22⋊Q8 — C5×C23.48D4
C1C2C2×C4 — C5×C23.48D4
C1C2×C10C22×C20 — C5×C23.48D4

Generators and relations for C5×C23.48D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 178 in 104 conjugacy classes, 54 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, C20 [×2], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C40 [×2], C2×C20 [×2], C2×C20 [×8], C5×Q8 [×2], C22×C10, C23.48D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], C22×C20, C22×C20, Q8×C10, C5×C22⋊C8, C5×Q8⋊C4 [×2], C5×C2.D8 [×2], C10×C4⋊C4, C5×C22⋊Q8, C5×C23.48D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], Q16 [×2], C2×D4, C4○D4 [×2], C2×C10 [×7], C22.D4, C2×Q16, C8⋊C22, C5×D4 [×2], C22×C10, C23.48D4, C5×Q16 [×2], D4×C10, C5×C4○D4 [×2], C5×C22.D4, C10×Q16, C5×C8⋊C22, C5×C23.48D4

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

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4G4H4I5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20H20I···20AB20AC···20AJ40A···40P
order122222444···4445555888810···1010···1020···2020···2020···2040···40
size111122224···488111144441···12···22···24···48···84···4

95 irreducible representations

dim1111111111112222222244
type++++++++-+
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4C4○D4Q16C5×D4C5×D4C5×C4○D4C5×Q16C8⋊C22C5×C8⋊C22
kernelC5×C23.48D4C5×C22⋊C8C5×Q8⋊C4C5×C2.D8C10×C4⋊C4C5×C22⋊Q8C23.48D4C22⋊C8Q8⋊C4C2.D8C2×C4⋊C4C22⋊Q8C2×C20C22×C10C20C2×C10C2×C4C23C4C22C10C2
# reps112211448844114444161614

Matrix representation of C5×C23.48D4 in GL4(𝔽41) generated by

18000
01800
0010
0001
,
40000
04000
0010
002240
,
1000
0100
00400
00040
,
40000
04000
0010
0001
,
291200
292900
00718
001134
,
203800
382100
00192
002522
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,22,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[29,29,0,0,12,29,0,0,0,0,7,11,0,0,18,34],[20,38,0,0,38,21,0,0,0,0,19,25,0,0,2,22] >;

C5×C23.48D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{48}D_4
% in TeX

G:=Group("C5xC2^3.48D4");
// GroupNames label

G:=SmallGroup(320,985);
// by ID

G=gap.SmallGroup(320,985);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,1066,10085,2539,124]);
// Polycyclic

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

׿
×
𝔽