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G = C24.13D10order 320 = 26·5

13rd non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.13D10, (C2×C20)⋊21D4, C10.95(C4×D4), (C2×Dic5)⋊10D4, C23.18(C4×D5), C2.1(C20⋊D4), (C22×C4).32D10, C22.102(D4×D5), C2.7(D10⋊D4), C10.85(C4⋊D4), C10.12(C41D4), Dic52(C22⋊C4), C2.3(Dic5⋊D4), C10.34(C4.4D4), C22.54(C4○D20), (C23×C10).40C22, (C22×C20).25C22, C53(C24.3C22), (C23×D5).13C22, C23.284(C22×D5), C2.28(Dic54D4), C22.49(D42D5), (C22×C10).331C23, C2.7(Dic5.5D4), (C22×Dic5).44C22, (C2×C5⋊D4)⋊10C4, (C2×C4)⋊9(C5⋊D4), (C2×C22⋊C4)⋊4D5, (C2×C4×Dic5)⋊24C2, C2.10(C4×C5⋊D4), (C2×C23.D5)⋊4C2, C2.30(D5×C22⋊C4), (C2×D10⋊C4)⋊5C2, (C10×C22⋊C4)⋊23C2, C22.128(C2×C4×D5), (C2×C10).323(C2×D4), C10.70(C2×C22⋊C4), (C22×C5⋊D4).3C2, C22.52(C2×C5⋊D4), (C2×C10.D4)⋊11C2, (C22×D5).25(C2×C4), (C2×C10).146(C4○D4), (C2×C10).211(C22×C4), (C22×C10).122(C2×C4), (C2×Dic5).105(C2×C4), SmallGroup(320,584)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.13D10
C1C5C10C2×C10C22×C10C23×D5C22×C5⋊D4 — C24.13D10
C5C2×C10 — C24.13D10
C1C23C2×C22⋊C4

Generators and relations for C24.13D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de9 >

Subgroups: 1022 in 258 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×4], C4 [×10], C22 [×7], C22 [×20], C5, C2×C4 [×2], C2×C4 [×18], D4 [×8], C23, C23 [×2], C23 [×14], D5 [×2], C10 [×7], C10 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×8], C24, C24, Dic5 [×4], Dic5 [×3], C20 [×3], D10 [×10], C2×C10 [×7], C2×C10 [×10], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×3], C2×C4⋊C4, C22×D4, C2×Dic5 [×8], C2×Dic5 [×5], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×5], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.3C22, C4×Dic5 [×2], C10.D4 [×2], D10⋊C4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×3], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, C2×C4×Dic5, C2×C10.D4, C2×D10⋊C4 [×2], C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C24.13D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C24.3C22, C2×C4×D5, C4○D20, D4×D5 [×3], D42D5, C2×C5⋊D4, D5×C22⋊C4, Dic54D4, D10⋊D4, Dic5.5D4, C4×C5⋊D4, Dic5⋊D4, C20⋊D4, C24.13D10

Smallest permutation representation of C24.13D10
On 160 points
Generators in S160
(1 96)(2 127)(3 98)(4 129)(5 100)(6 131)(7 82)(8 133)(9 84)(10 135)(11 86)(12 137)(13 88)(14 139)(15 90)(16 121)(17 92)(18 123)(19 94)(20 125)(21 89)(22 140)(23 91)(24 122)(25 93)(26 124)(27 95)(28 126)(29 97)(30 128)(31 99)(32 130)(33 81)(34 132)(35 83)(36 134)(37 85)(38 136)(39 87)(40 138)(41 118)(42 68)(43 120)(44 70)(45 102)(46 72)(47 104)(48 74)(49 106)(50 76)(51 108)(52 78)(53 110)(54 80)(55 112)(56 62)(57 114)(58 64)(59 116)(60 66)(61 160)(63 142)(65 144)(67 146)(69 148)(71 150)(73 152)(75 154)(77 156)(79 158)(101 149)(103 151)(105 153)(107 155)(109 157)(111 159)(113 141)(115 143)(117 145)(119 147)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 61)(40 62)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 121)(60 122)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(41 146)(42 147)(43 148)(44 149)(45 150)(46 151)(47 152)(48 153)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 141)(57 142)(58 143)(59 144)(60 145)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)
(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 10 38 27)(2 26 39 9)(3 8 40 25)(4 24 21 7)(5 6 22 23)(11 20 28 37)(12 36 29 19)(13 18 30 35)(14 34 31 17)(15 16 32 33)(41 98 156 138)(42 137 157 97)(43 96 158 136)(44 135 159 95)(45 94 160 134)(46 133 141 93)(47 92 142 132)(48 131 143 91)(49 90 144 130)(50 129 145 89)(51 88 146 128)(52 127 147 87)(53 86 148 126)(54 125 149 85)(55 84 150 124)(56 123 151 83)(57 82 152 122)(58 121 153 81)(59 100 154 140)(60 139 155 99)(61 109 102 68)(62 67 103 108)(63 107 104 66)(64 65 105 106)(69 101 110 80)(70 79 111 120)(71 119 112 78)(72 77 113 118)(73 117 114 76)(74 75 115 116)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G···4N4O4P5A5B10A···10N10O···10V20A···20P
order12···222224444444···4445510···1010···1020···20
size11···144202022224410···102020222···24···44···4

68 irreducible representations

dim1111111122222222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C4D4D4D5C4○D4D10D10C5⋊D4C4×D5C4○D20D4×D5D42D5
kernelC24.13D10C2×C4×Dic5C2×C10.D4C2×D10⋊C4C2×C23.D5C10×C22⋊C4C22×C5⋊D4C2×C5⋊D4C2×Dic5C2×C20C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22C22
# reps1112111862244288862

Matrix representation of C24.13D10 in GL5(𝔽41)

400000
023100
051800
000400
000401
,
400000
040000
004000
000400
000040
,
10000
040000
004000
00010
00001
,
10000
01000
00100
000400
000040
,
320000
002200
0281300
0003218
00009
,
90000
0282200
0371300
000923
000932

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,23,5,0,0,0,1,18,0,0,0,0,0,40,40,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,0,28,0,0,0,22,13,0,0,0,0,0,32,0,0,0,0,18,9],[9,0,0,0,0,0,28,37,0,0,0,22,13,0,0,0,0,0,9,9,0,0,0,23,32] >;

C24.13D10 in GAP, Magma, Sage, TeX

C_2^4._{13}D_{10}
% in TeX

G:=Group("C2^4.13D10");
// GroupNames label

G:=SmallGroup(320,584);
// by ID

G=gap.SmallGroup(320,584);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,f^2=d*b=b*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*e=e*b,b*f=f*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=d*e^9>;
// generators/relations

׿
×
𝔽