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## G = C4⋊C4.178D10order 320 = 26·5

### 51st non-split extension by C4⋊C4 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊C4.178D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C4⋊C4.178D10
 Lower central C5 — C2×C10 — C4⋊C4.178D10
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C4⋊C4.178D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 670 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C23 [×2], C10 [×3], C10 [×4], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×C10, C22×C10 [×2], C23.36C23, C4×Dic5 [×4], C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×2], C23.D5 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, D4×C10, D4×C10 [×2], C23.11D10 [×2], C23.D10 [×2], Dic53Q8, C4.Dic10, C2×C4×Dic5, C20.48D4, D4×Dic5, D4×Dic5 [×2], C23.18D10 [×2], C20.17D4, C5×C4⋊D4, C4⋊C4.178D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, D42D5 [×4], C23×D5, C2×D42D5 [×2], D5×C4○D4, C4⋊C4.178D10

Smallest permutation representation of C4⋊C4.178D10
On 160 points
Generators in S160
```(1 23 32 46)(2 24 33 47)(3 25 34 48)(4 26 35 49)(5 27 36 50)(6 28 37 41)(7 29 38 42)(8 30 39 43)(9 21 40 44)(10 22 31 45)(11 74 54 112)(12 75 55 113)(13 76 56 114)(14 77 57 115)(15 78 58 116)(16 79 59 117)(17 80 60 118)(18 71 51 119)(19 72 52 120)(20 73 53 111)(61 96 158 148)(62 97 159 149)(63 98 160 150)(64 99 151 141)(65 100 152 142)(66 91 153 143)(67 92 154 144)(68 93 155 145)(69 94 156 146)(70 95 157 147)(81 123 133 106)(82 124 134 107)(83 125 135 108)(84 126 136 109)(85 127 137 110)(86 128 138 101)(87 129 139 102)(88 130 140 103)(89 121 131 104)(90 122 132 105)
(1 116 111 6)(2 7 112 117)(3 118 113 8)(4 9 114 119)(5 120 115 10)(11 59 24 42)(12 43 25 60)(13 51 26 44)(14 45 27 52)(15 53 28 46)(16 47 29 54)(17 55 30 48)(18 49 21 56)(19 57 22 50)(20 41 23 58)(31 36 72 77)(32 78 73 37)(33 38 74 79)(34 80 75 39)(35 40 76 71)(61 66 108 103)(62 104 109 67)(63 68 110 105)(64 106 101 69)(65 70 102 107)(81 138 94 141)(82 142 95 139)(83 140 96 143)(84 144 97 131)(85 132 98 145)(86 146 99 133)(87 134 100 147)(88 148 91 135)(89 136 92 149)(90 150 93 137)(121 126 154 159)(122 160 155 127)(123 128 156 151)(124 152 157 129)(125 130 158 153)
(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)
(1 156 73 106)(2 155 74 105)(3 154 75 104)(4 153 76 103)(5 152 77 102)(6 151 78 101)(7 160 79 110)(8 159 80 109)(9 158 71 108)(10 157 72 107)(11 132 47 93)(12 131 48 92)(13 140 49 91)(14 139 50 100)(15 138 41 99)(16 137 42 98)(17 136 43 97)(18 135 44 96)(19 134 45 95)(20 133 46 94)(21 148 51 83)(22 147 52 82)(23 146 53 81)(24 145 54 90)(25 144 55 89)(26 143 56 88)(27 142 57 87)(28 141 58 86)(29 150 59 85)(30 149 60 84)(31 70 120 124)(32 69 111 123)(33 68 112 122)(34 67 113 121)(35 66 114 130)(36 65 115 129)(37 64 116 128)(38 63 117 127)(39 62 118 126)(40 61 119 125)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 C4○D4 D10 D10 D10 D10 D4⋊2D5 D4⋊2D5 D5×C4○D4 kernel C4⋊C4.178D10 C23.11D10 C23.D10 Dic5⋊3Q8 C4.Dic10 C2×C4×Dic5 C20.48D4 D4×Dic5 C23.18D10 C20.17D4 C5×C4⋊D4 C4⋊D4 Dic5 C20 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C22 C2 # reps 1 2 2 1 1 1 1 3 2 1 1 2 4 4 4 4 2 2 6 4 4 4

Matrix representation of C4⋊C4.178D10 in GL6(𝔽41)

 1 5 0 0 0 0 16 40 0 0 0 0 0 0 1 25 0 0 0 0 36 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 4 0 0 0 0 0 32 0 0 0 0 0 0 1 25 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 4 0 0 0 0 21 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 6 0 0 0 0 34 0
,
 32 37 0 0 0 0 20 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 8 1

`G:=sub<GL(6,GF(41))| [1,16,0,0,0,0,5,40,0,0,0,0,0,0,1,36,0,0,0,0,25,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,4,32,0,0,0,0,0,0,1,0,0,0,0,0,25,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,21,0,0,0,0,4,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,6,0],[32,20,0,0,0,0,37,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,8,0,0,0,0,0,1] >;`

C4⋊C4.178D10 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{178}D_{10}`
`% in TeX`

`G:=Group("C4:C4.178D10");`
`// GroupNames label`

`G:=SmallGroup(320,1272);`
`// by ID`

`G=gap.SmallGroup(320,1272);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,794,297,12550]);`
`// Polycyclic`

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

׿
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