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## G = C42.276D10order 320 = 26·5

### 35th non-split extension by C42 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.276D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C4×D20 — C42.276D10
 Lower central C5 — C2×C10 — C42.276D10
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C42.276D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >

Subgroups: 1182 in 310 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C42, C4×D4, C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C22.26C24, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, C202Q8, C4×D20, C204D4, C4.D20, C207D4, C2×C4×C20, C2×C4○D20, C42.276D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, D20, C22×D5, C22.26C24, C2×D20, C4○D20, C23×D5, C22×D20, C2×C4○D20, C42.276D10

Smallest permutation representation of C42.276D10
On 160 points
Generators in S160
```(1 113 33 94)(2 114 34 95)(3 115 35 96)(4 116 36 97)(5 117 37 98)(6 118 38 99)(7 119 39 100)(8 120 40 91)(9 111 31 92)(10 112 32 93)(11 125 45 107)(12 126 46 108)(13 127 47 109)(14 128 48 110)(15 129 49 101)(16 130 50 102)(17 121 41 103)(18 122 42 104)(19 123 43 105)(20 124 44 106)(21 85 147 67)(22 86 148 68)(23 87 149 69)(24 88 150 70)(25 89 141 61)(26 90 142 62)(27 81 143 63)(28 82 144 64)(29 83 145 65)(30 84 146 66)(51 160 80 131)(52 151 71 132)(53 152 72 133)(54 153 73 134)(55 154 74 135)(56 155 75 136)(57 156 76 137)(58 157 77 138)(59 158 78 139)(60 159 79 140)
(1 62 16 54)(2 63 17 55)(3 64 18 56)(4 65 19 57)(5 66 20 58)(6 67 11 59)(7 68 12 60)(8 69 13 51)(9 70 14 52)(10 61 15 53)(21 125 158 118)(22 126 159 119)(23 127 160 120)(24 128 151 111)(25 129 152 112)(26 130 153 113)(27 121 154 114)(28 122 155 115)(29 123 156 116)(30 124 157 117)(31 88 48 71)(32 89 49 72)(33 90 50 73)(34 81 41 74)(35 82 42 75)(36 83 43 76)(37 84 44 77)(38 85 45 78)(39 86 46 79)(40 87 47 80)(91 149 109 131)(92 150 110 132)(93 141 101 133)(94 142 102 134)(95 143 103 135)(96 144 104 136)(97 145 105 137)(98 146 106 138)(99 147 107 139)(100 148 108 140)
(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 93 16 101)(2 110 17 92)(3 91 18 109)(4 108 19 100)(5 99 20 107)(6 106 11 98)(7 97 12 105)(8 104 13 96)(9 95 14 103)(10 102 15 94)(21 77 158 84)(22 83 159 76)(23 75 160 82)(24 81 151 74)(25 73 152 90)(26 89 153 72)(27 71 154 88)(28 87 155 80)(29 79 156 86)(30 85 157 78)(31 114 48 121)(32 130 49 113)(33 112 50 129)(34 128 41 111)(35 120 42 127)(36 126 43 119)(37 118 44 125)(38 124 45 117)(39 116 46 123)(40 122 47 115)(51 144 69 136)(52 135 70 143)(53 142 61 134)(54 133 62 141)(55 150 63 132)(56 131 64 149)(57 148 65 140)(58 139 66 147)(59 146 67 138)(60 137 68 145)```

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 20 20 20 20 1 1 1 1 2 ··· 2 20 20 20 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D20 C4○D20 kernel C42.276D10 C20⋊2Q8 C4×D20 C20⋊4D4 C4.D20 C20⋊7D4 C2×C4×C20 C2×C4○D20 C2×C20 C2×C42 C20 C42 C22×C4 C2×C4 C4 # reps 1 1 4 1 2 4 1 2 4 2 8 8 6 16 32

Matrix representation of C42.276D10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 30 28 0 0 22 11
,
 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
,
 1 37 0 0 21 38 0 0 0 0 3 21 0 0 37 40
,
 38 4 0 0 18 3 0 0 0 0 19 32 0 0 22 22
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,30,22,0,0,28,11],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[1,21,0,0,37,38,0,0,0,0,3,37,0,0,21,40],[38,18,0,0,4,3,0,0,0,0,19,22,0,0,32,22] >;`

C42.276D10 in GAP, Magma, Sage, TeX

`C_4^2._{276}D_{10}`
`% in TeX`

`G:=Group("C4^2.276D10");`
`// GroupNames label`

`G:=SmallGroup(320,1149);`
`// by ID`

`G=gap.SmallGroup(320,1149);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,12550]);`
`// Polycyclic`

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

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