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## G = C42.228D10order 320 = 26·5

### 48th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.228D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.228D10
 Lower central C5 — C2×C10 — C42.228D10
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C42.228D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >

Subgroups: 1150 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×10], C22, C22 [×16], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×21], D4 [×20], Q8 [×4], C23 [×2], C23 [×3], D5 [×4], C10 [×3], C10 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4 [×9], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], Dic5 [×3], C20 [×4], C20 [×3], D10 [×2], D10 [×8], C2×C10, C2×C10 [×6], C2×C42, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], C4×D5 [×8], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×12], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5, C22×D5 [×2], C22×C10 [×2], C22.26C24, C4×Dic5 [×3], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×4], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×3], C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C4○D20 [×8], C2×C5⋊D4 [×6], C22×C20 [×2], D4×C10, D5×C42, C4×D20, D10⋊D4 [×2], Dic5.5D4 [×2], C20⋊Q8, C4⋊D20, C4×C5⋊D4 [×2], C202D4, C20⋊D4, D4×C20, C2×C4○D20 [×2], C42.228D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D5×C4○D4, C42.228D10

Smallest permutation representation of C42.228D10
On 160 points
Generators in S160
```(1 108 33 113)(2 114 34 109)(3 110 35 115)(4 116 36 101)(5 102 37 117)(6 118 38 103)(7 104 39 119)(8 120 40 105)(9 106 31 111)(10 112 32 107)(11 98 48 123)(12 124 49 99)(13 100 50 125)(14 126 41 91)(15 92 42 127)(16 128 43 93)(17 94 44 129)(18 130 45 95)(19 96 46 121)(20 122 47 97)(21 90 135 58)(22 59 136 81)(23 82 137 60)(24 51 138 83)(25 84 139 52)(26 53 140 85)(27 86 131 54)(28 55 132 87)(29 88 133 56)(30 57 134 89)(61 142 77 157)(62 158 78 143)(63 144 79 159)(64 160 80 145)(65 146 71 151)(66 152 72 147)(67 148 73 153)(68 154 74 149)(69 150 75 155)(70 156 76 141)
(1 53 13 67)(2 54 14 68)(3 55 15 69)(4 56 16 70)(5 57 17 61)(6 58 18 62)(7 59 19 63)(8 60 20 64)(9 51 11 65)(10 52 12 66)(21 130 158 118)(22 121 159 119)(23 122 160 120)(24 123 151 111)(25 124 152 112)(26 125 153 113)(27 126 154 114)(28 127 155 115)(29 128 156 116)(30 129 157 117)(31 83 48 71)(32 84 49 72)(33 85 50 73)(34 86 41 74)(35 87 42 75)(36 88 43 76)(37 89 44 77)(38 90 45 78)(39 81 46 79)(40 82 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)
(1 142 13 134)(2 133 14 141)(3 150 15 132)(4 131 16 149)(5 148 17 140)(6 139 18 147)(7 146 19 138)(8 137 20 145)(9 144 11 136)(10 135 12 143)(21 49 158 32)(22 31 159 48)(23 47 160 40)(24 39 151 46)(25 45 152 38)(26 37 153 44)(27 43 154 36)(28 35 155 42)(29 41 156 34)(30 33 157 50)(51 104 65 96)(52 95 66 103)(53 102 67 94)(54 93 68 101)(55 110 69 92)(56 91 70 109)(57 108 61 100)(58 99 62 107)(59 106 63 98)(60 97 64 105)(71 121 83 119)(72 118 84 130)(73 129 85 117)(74 116 86 128)(75 127 87 115)(76 114 88 126)(77 125 89 113)(78 112 90 124)(79 123 81 111)(80 120 82 122)```

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K ··· 4P 4Q 4R 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 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 4 4 10 10 20 20 1 1 1 1 2 2 2 2 4 4 10 ··· 10 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 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 D4 D5 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4○D20 D4×D5 D5×C4○D4 kernel C42.228D10 D5×C42 C4×D20 D10⋊D4 Dic5.5D4 C20⋊Q8 C4⋊D20 C4×C5⋊D4 C20⋊2D4 C20⋊D4 D4×C20 C2×C4○D20 C4×D5 C4×D4 Dic5 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C4 C2 # reps 1 1 1 2 2 1 1 2 1 1 1 2 4 2 4 4 2 4 2 4 2 16 4 4

Matrix representation of C42.228D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 34 0 0 0 0 6 35 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 9 0 0 0 0 0 0 32 0 0 0 0 0 0 6 34 0 0 0 0 5 35 0 0 0 0 0 0 0 40 0 0 0 0 40 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,34,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,6,5,0,0,0,0,34,35,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;`

C42.228D10 in GAP, Magma, Sage, TeX

`C_4^2._{228}D_{10}`
`% in TeX`

`G:=Group("C4^2.228D10");`
`// GroupNames label`

`G:=SmallGroup(320,1220);`
`// by ID`

`G=gap.SmallGroup(320,1220);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,570,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,c*a*c^-1=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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