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## G = C42.178D10order 320 = 26·5

### 178th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.178D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊3Q8 — C42.178D10
 Lower central C5 — C2×C10 — C42.178D10
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.178D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 798 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×4], Q8 [×4], C23 [×3], D5 [×3], C10 [×3], C42, C42 [×3], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8 [×2], C2×Q8, Dic5 [×5], C20 [×2], C20 [×6], D10 [×9], C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C422C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22.36C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q8×C10 [×2], C42⋊D5, C4.D20, Dic53Q8, D208C4, D10.13D4 [×2], C4⋊D20, D102Q8, C4⋊C4⋊D5 [×2], D103Q8 [×2], C20.23D4 [×2], C5×C4⋊Q8, C42.178D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.36C24, Q82D5 [×2], C23×D5, D46D10, C2×Q82D5, Q8.10D10, C42.178D10

Smallest permutation representation of C42.178D10
On 160 points
Generators in S160
```(1 57 11 47)(2 48 12 58)(3 59 13 49)(4 50 14 60)(5 41 15 51)(6 52 16 42)(7 43 17 53)(8 54 18 44)(9 45 19 55)(10 56 20 46)(21 83 31 93)(22 94 32 84)(23 85 33 95)(24 96 34 86)(25 87 35 97)(26 98 36 88)(27 89 37 99)(28 100 38 90)(29 91 39 81)(30 82 40 92)(61 117 71 107)(62 108 72 118)(63 119 73 109)(64 110 74 120)(65 101 75 111)(66 112 76 102)(67 103 77 113)(68 114 78 104)(69 105 79 115)(70 116 80 106)(121 150 131 160)(122 141 132 151)(123 152 133 142)(124 143 134 153)(125 154 135 144)(126 145 136 155)(127 156 137 146)(128 147 138 157)(129 158 139 148)(130 149 140 159)
(1 75 138 81)(2 82 139 76)(3 77 140 83)(4 84 121 78)(5 79 122 85)(6 86 123 80)(7 61 124 87)(8 88 125 62)(9 63 126 89)(10 90 127 64)(11 65 128 91)(12 92 129 66)(13 67 130 93)(14 94 131 68)(15 69 132 95)(16 96 133 70)(17 71 134 97)(18 98 135 72)(19 73 136 99)(20 100 137 74)(21 49 103 149)(22 150 104 50)(23 51 105 151)(24 152 106 52)(25 53 107 153)(26 154 108 54)(27 55 109 155)(28 156 110 56)(29 57 111 157)(30 158 112 58)(31 59 113 159)(32 160 114 60)(33 41 115 141)(34 142 116 42)(35 43 117 143)(36 144 118 44)(37 45 119 145)(38 146 120 46)(39 47 101 147)(40 148 102 48)
(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 85 11 95)(2 94 12 84)(3 83 13 93)(4 92 14 82)(5 81 15 91)(6 90 16 100)(7 99 17 89)(8 88 18 98)(9 97 19 87)(10 86 20 96)(21 149 31 159)(22 158 32 148)(23 147 33 157)(24 156 34 146)(25 145 35 155)(26 154 36 144)(27 143 37 153)(28 152 38 142)(29 141 39 151)(30 150 40 160)(41 101 51 111)(42 110 52 120)(43 119 53 109)(44 108 54 118)(45 117 55 107)(46 106 56 116)(47 115 57 105)(48 104 58 114)(49 113 59 103)(50 102 60 112)(61 126 71 136)(62 135 72 125)(63 124 73 134)(64 133 74 123)(65 122 75 132)(66 131 76 121)(67 140 77 130)(68 129 78 139)(69 138 79 128)(70 127 80 137)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4H 4I 4J 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 20 20 20 2 2 4 ··· 4 10 10 10 10 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2+ 1+4 2- 1+4 Q8⋊2D5 D4⋊6D10 Q8.10D10 kernel C42.178D10 C42⋊D5 C4.D20 Dic5⋊3Q8 D20⋊8C4 D10.13D4 C4⋊D20 D10⋊2Q8 C4⋊C4⋊D5 D10⋊3Q8 C20.23D4 C5×C4⋊Q8 C4⋊Q8 C20 C42 C4⋊C4 C2×Q8 C10 C10 C4 C2 C2 # reps 1 1 1 1 1 2 1 1 2 2 2 1 2 4 2 8 4 1 1 4 4 4

Matrix representation of C42.178D10 in GL8(𝔽41)

 0 0 40 1 0 0 0 0 7 1 39 34 0 0 0 0 38 23 40 0 0 0 0 0 37 23 40 0 0 0 0 0 0 0 0 0 23 1 0 38 0 0 0 0 5 18 3 38 0 0 0 0 28 13 24 40 0 0 0 0 28 0 1 17
,
 0 0 40 1 0 0 0 0 7 1 39 34 0 0 0 0 38 23 40 0 0 0 0 0 37 23 40 0 0 0 0 0 0 0 0 0 23 1 0 0 0 0 0 0 5 18 0 0 0 0 0 0 0 0 17 1 0 0 0 0 0 0 40 24
,
 15 12 0 2 0 0 0 0 31 39 39 29 0 0 0 0 24 0 12 29 0 0 0 0 30 24 12 16 0 0 0 0 0 0 0 0 0 12 9 9 0 0 0 0 19 22 19 18 0 0 0 0 31 29 31 29 0 0 0 0 3 19 12 29
,
 33 34 8 14 0 0 0 0 40 35 7 6 0 0 0 0 8 6 7 7 0 0 0 0 2 13 7 7 0 0 0 0 0 0 0 0 21 3 0 0 0 0 0 0 3 20 0 0 0 0 0 0 0 0 24 40 0 0 0 0 0 0 3 17

`G:=sub<GL(8,GF(41))| [0,7,38,37,0,0,0,0,0,1,23,23,0,0,0,0,40,39,40,40,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,23,5,28,28,0,0,0,0,1,18,13,0,0,0,0,0,0,3,24,1,0,0,0,0,38,38,40,17],[0,7,38,37,0,0,0,0,0,1,23,23,0,0,0,0,40,39,40,40,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,23,5,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,17,40,0,0,0,0,0,0,1,24],[15,31,24,30,0,0,0,0,12,39,0,24,0,0,0,0,0,39,12,12,0,0,0,0,2,29,29,16,0,0,0,0,0,0,0,0,0,19,31,3,0,0,0,0,12,22,29,19,0,0,0,0,9,19,31,12,0,0,0,0,9,18,29,29],[33,40,8,2,0,0,0,0,34,35,6,13,0,0,0,0,8,7,7,7,0,0,0,0,14,6,7,7,0,0,0,0,0,0,0,0,21,3,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,24,3,0,0,0,0,0,0,40,17] >;`

C42.178D10 in GAP, Magma, Sage, TeX

`C_4^2._{178}D_{10}`
`% in TeX`

`G:=Group("C4^2.178D10");`
`// GroupNames label`

`G:=SmallGroup(320,1405);`
`// by ID`

`G=gap.SmallGroup(320,1405);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,675,570,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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