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## G = C42.7D10order 320 = 26·5

### 7th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.7D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×C20 — C42.D5 — C42.7D10
 Lower central C5 — C2×C10 — C2×C20 — C42.7D10
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.7D10
G = < a,b,c,d | a4=b4=c10=1, d2=ab-1, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc-1 >

Subgroups: 206 in 70 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, C20, C2×C10, C2×C10, C8⋊C4, C4.4D4, C52C8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.C22, C2×C52C8, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C42.D5, C5×C4.4D4, C42.7D10
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C4.D4, C4≀C2, C2×Dic5, C5⋊D4, C42.C22, C23.D5, C20.D4, D42Dic5, C42.7D10

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + - - + image C1 C2 C2 C4 C4 D4 D5 D10 Dic5 Dic5 C4≀C2 C5⋊D4 C4.D4 C20.D4 D4⋊2Dic5 kernel C42.7D10 C42.D5 C5×C4.4D4 D4×C10 Q8×C10 C2×C20 C4.4D4 C42 C2×D4 C2×Q8 C10 C2×C4 C10 C2 C2 # reps 1 2 1 2 2 2 2 2 2 2 8 8 1 4 8

Matrix representation of C42.7D10 in GL6(𝔽41)

 0 32 0 0 0 0 9 0 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 0 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40
,
 4 4 0 0 0 0 4 37 0 0 0 0 0 0 5 36 0 0 0 0 5 5 0 0 0 0 0 0 4 27 0 0 0 0 10 37

`G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[4,4,0,0,0,0,4,37,0,0,0,0,0,0,5,5,0,0,0,0,36,5,0,0,0,0,0,0,4,10,0,0,0,0,27,37] >;`

C42.7D10 in GAP, Magma, Sage, TeX

`C_4^2._7D_{10}`
`% in TeX`

`G:=Group("C4^2.7D10");`
`// GroupNames label`

`G:=SmallGroup(320,98);`
`// by ID`

`G=gap.SmallGroup(320,98);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,268,1571,570,136,12550]);`
`// Polycyclic`

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

׿
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