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## G = C42.102D10order 320 = 26·5

### 102nd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.102D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C42.102D10
 Lower central C5 — C2×C10 — C42.102D10
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C42.102D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 718 in 234 conjugacy classes, 101 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×5], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, D5, C10 [×3], C10 [×3], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×4], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×7], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5, C22×C10 [×2], C23.36C23, C4×Dic5 [×5], C10.D4 [×6], C4⋊Dic5 [×3], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C42⋊D5, Dic5.14D4, C23.D10, Dic5.5D4, C22.D20, Dic5.Q8, C4⋊C4⋊D5, C2×C4×Dic5, C23.21D10, C4×C5⋊D4 [×2], C23.18D10, Dic5⋊D4, D4×C20, C42.102D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C4○D20 [×2], D42D5 [×2], C23×D5, C2×C4○D20, C2×D42D5, D5×C4○D4, C42.102D10

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4○D20 D4⋊2D5 D5×C4○D4 kernel C42.102D10 C4×Dic10 C42⋊D5 Dic5.14D4 C23.D10 Dic5.5D4 C22.D20 Dic5.Q8 C4⋊C4⋊D5 C2×C4×Dic5 C23.21D10 C4×C5⋊D4 C23.18D10 Dic5⋊D4 D4×C20 C4×D4 Dic5 C20 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C4 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 4 4 4 2 4 2 4 2 16 4 4

Matrix representation of C42.102D10 in GL4(𝔽41) generated by

 23 6 0 0 35 18 0 0 0 0 32 0 0 0 32 9
,
 32 0 0 0 0 32 0 0 0 0 40 0 0 0 0 40
,
 35 35 0 0 6 40 0 0 0 0 21 40 0 0 30 20
,
 21 20 0 0 23 20 0 0 0 0 9 0 0 0 0 9
`G:=sub<GL(4,GF(41))| [23,35,0,0,6,18,0,0,0,0,32,32,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[35,6,0,0,35,40,0,0,0,0,21,30,0,0,40,20],[21,23,0,0,20,20,0,0,0,0,9,0,0,0,0,9] >;`

C42.102D10 in GAP, Magma, Sage, TeX

`C_4^2._{102}D_{10}`
`% in TeX`

`G:=Group("C4^2.102D10");`
`// GroupNames label`

`G:=SmallGroup(320,1210);`
`// by ID`

`G=gap.SmallGroup(320,1210);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1123,794,12550]);`
`// Polycyclic`

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

׿
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