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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23.(C2×F5)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — C23.(C2×F5)
 Lower central C5 — C2×C10 — C23.(C2×F5)
 Upper central C1 — C22 — C22⋊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, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C42.7C22, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C2×C5⋊C8, C22×Dic5, C4×C5⋊C8, C20⋊C8, C10.C42, Dic5⋊C8, C23.2F5, C23.11D10, C23.(C2×F5)
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, F5, C42⋊C2, C8○D4, C2×F5, C42.7C22, C22×F5, D10.C23, D4.F5, C23.(C2×F5)

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 8 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4○D4 C8○D4 F5 C2×F5 C2×F5 D10.C23 D4.F5 kernel C23.(C2×F5) C4×C5⋊C8 C20⋊C8 C10.C42 Dic5⋊C8 C23.2F5 C23.11D10 C10.D4 C23.D5 C5×C22⋊C4 Dic5 C10 C22⋊C4 C2×C4 C23 C2 C2 # reps 1 1 1 1 1 2 1 4 2 2 4 8 1 2 1 4 2

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

 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 40 0 0 0 0 8 40 0 0 0 0 0 0 0 35 0 0 0 0 7 34
,
 14 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 1 0 0 0 0 0 1 0 0

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

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