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G = C23.(C2×F5)  order 320 = 26·5

5th non-split extension by C23 of C2×F5 acting via C2×F5/D5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20⋊C812C2, C22⋊C4.6F5, C23.9(C2×F5), C10.7(C8○D4), C23.D5.4C4, Dic5⋊C89C2, C10.D4.8C4, C2.10(D4.F5), C10.C4212C2, C10.9(C42⋊C2), C23.2F5.3C2, Dic5.27(C4○D4), C22.73(C22×F5), C51(C42.7C22), (C2×Dic5).327C23, (C4×Dic5).243C22, C23.11D10.7C2, (C22×Dic5).182C22, C2.12(D10.C23), (C4×C5⋊C8)⋊13C2, (C2×C4).58(C2×F5), (C2×C20).82(C2×C4), (C5×C22⋊C4).6C4, (C2×C5⋊C8).24C22, (C22×C10).18(C2×C4), (C2×C10).35(C22×C4), (C2×Dic5).52(C2×C4), SmallGroup(320,1035)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C23.(C2×F5)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C4×C5⋊C8 — C23.(C2×F5)
C5C2×C10 — C23.(C2×F5)
C1C22C22⋊C4

Generators and relations for C23.(C2×F5)
 G = < a,b,c,d,e | a2=b2=c4=d5=1, e4=b, cac-1=ab=ba, ad=da, eae-1=ac2, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 298 in 96 conjugacy classes, 42 normal (all characteristic)
C1, C2 [×3], C2, C4 [×7], C22, C22 [×3], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C23, C10 [×3], C10, C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×3], C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C42⋊C2, C5⋊C8 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×C10, C42.7C22, C4×Dic5 [×2], C10.D4 [×2], C23.D5, C5×C22⋊C4, C2×C5⋊C8 [×4], C22×Dic5, C4×C5⋊C8, C20⋊C8, C10.C42, Dic5⋊C8, C23.2F5 [×2], C23.11D10, C23.(C2×F5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], F5, C42⋊C2, C8○D4 [×2], C2×F5 [×3], C42.7C22, C22×F5, D10.C23, D4.F5 [×2], C23.(C2×F5)

Smallest permutation representation of C23.(C2×F5)
On 160 points
Generators in S160
(1 137)(3 139)(5 141)(7 143)(9 134)(11 136)(13 130)(15 132)(17 21)(18 40)(19 23)(20 34)(22 36)(24 38)(25 147)(26 30)(27 149)(28 32)(29 151)(31 145)(33 37)(35 39)(41 45)(42 100)(43 47)(44 102)(46 104)(48 98)(49 53)(50 111)(51 55)(52 105)(54 107)(56 109)(57 125)(59 127)(61 121)(63 123)(65 87)(66 70)(67 81)(68 72)(69 83)(71 85)(73 92)(75 94)(77 96)(79 90)(82 86)(84 88)(97 101)(99 103)(106 110)(108 112)(113 159)(115 153)(117 155)(119 157)(146 150)(148 152)
(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 18 137 36)(2 19 138 37)(3 20 139 38)(4 21 140 39)(5 22 141 40)(6 23 142 33)(7 24 143 34)(8 17 144 35)(9 100 134 46)(10 101 135 47)(11 102 136 48)(12 103 129 41)(13 104 130 42)(14 97 131 43)(15 98 132 44)(16 99 133 45)(25 59 151 127)(26 60 152 128)(27 61 145 121)(28 62 146 122)(29 63 147 123)(30 64 148 124)(31 57 149 125)(32 58 150 126)(49 114 106 160)(50 115 107 153)(51 116 108 154)(52 117 109 155)(53 118 110 156)(54 119 111 157)(55 120 112 158)(56 113 105 159)(65 96 83 77)(66 89 84 78)(67 90 85 79)(68 91 86 80)(69 92 87 73)(70 93 88 74)(71 94 81 75)(72 95 82 76)
(1 94 132 63 113)(2 64 95 114 133)(3 115 57 134 96)(4 135 116 89 58)(5 90 136 59 117)(6 60 91 118 129)(7 119 61 130 92)(8 131 120 93 62)(9 77 139 153 125)(10 154 78 126 140)(11 127 155 141 79)(12 142 128 80 156)(13 73 143 157 121)(14 158 74 122 144)(15 123 159 137 75)(16 138 124 76 160)(17 43 112 88 146)(18 81 44 147 105)(19 148 82 106 45)(20 107 149 46 83)(21 47 108 84 150)(22 85 48 151 109)(23 152 86 110 41)(24 111 145 42 87)(25 52 40 67 102)(26 68 53 103 33)(27 104 69 34 54)(28 35 97 55 70)(29 56 36 71 98)(30 72 49 99 37)(31 100 65 38 50)(32 39 101 51 66)
(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)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K 5 8A···8H8I8J8K8L10A10B10C10D10E20A20B20C20D
order122224444444444458···88888101010101020202020
size11114224555510102020410···1020202020444888888

38 irreducible representations

dim11111111112244448
type++++++++++-
imageC1C2C2C2C2C2C2C4C4C4C4○D4C8○D4F5C2×F5C2×F5D10.C23D4.F5
kernelC23.(C2×F5)C4×C5⋊C8C20⋊C8C10.C42Dic5⋊C8C23.2F5C23.11D10C10.D4C23.D5C5×C22⋊C4Dic5C10C22⋊C4C2×C4C23C2C2
# reps11111214224812142

Matrix representation of C23.(C2×F5) in GL6(𝔽41)

100000
0400000
0040000
0004000
000010
000001
,
4000000
0400000
001000
000100
000010
000001
,
010000
100000
009000
000900
000090
000009
,
100000
010000
0074000
0084000
0000035
0000734
,
1400000
0140000
000010
000001
0040100
000100

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,40,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34],[14,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C23.(C2×F5) in GAP, Magma, Sage, TeX

C_2^3.(C_2\times F_5)
% in TeX

G:=Group("C2^3.(C2xF5)");
// GroupNames label

G:=SmallGroup(320,1035);
// by ID

G=gap.SmallGroup(320,1035);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,136,6278,1595]);
// Polycyclic

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

׿
×
𝔽