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## G = C42.232D10order 320 = 26·5

### 52nd non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.232D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.232D10
 Lower central C5 — C2×C10 — C42.232D10
 Upper central C1 — C2×C4 — C4×Q8

Generators and relations for C42.232D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c9 >

Subgroups: 670 in 222 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×14], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], Q8 [×8], C23, D5 [×2], C10 [×3], C42, C42 [×2], C42 [×5], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×3], C2×Q8, C2×Q8 [×3], Dic5 [×4], Dic5 [×5], C20 [×4], C20 [×5], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2 [×2], C4×Q8, C4×Q8 [×3], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], Dic10 [×6], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C23.37C23, C4×Dic5 [×3], C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×4], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×3], Q8×C10, C4×Dic10, C4×Dic10 [×2], D5×C42, C42⋊D5 [×2], C20⋊Q8, Dic5.Q8 [×2], D10⋊Q8 [×2], D102Q8, Dic5⋊Q8, D103Q8, Q8×C20, C42.232D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C22×Q8, C2×C4○D4 [×2], C22×D5 [×7], C23.37C23, C4○D20 [×2], Q8×D5 [×2], C23×D5, C2×C4○D20, C2×Q8×D5, D5×C4○D4, C42.232D10

Smallest permutation representation of C42.232D10
On 160 points
Generators in S160
```(1 116 11 106)(2 107 12 117)(3 118 13 108)(4 109 14 119)(5 120 15 110)(6 111 16 101)(7 102 17 112)(8 113 18 103)(9 104 19 114)(10 115 20 105)(21 146 31 156)(22 157 32 147)(23 148 33 158)(24 159 34 149)(25 150 35 160)(26 141 36 151)(27 152 37 142)(28 143 38 153)(29 154 39 144)(30 145 40 155)(41 71 51 61)(42 62 52 72)(43 73 53 63)(44 64 54 74)(45 75 55 65)(46 66 56 76)(47 77 57 67)(48 68 58 78)(49 79 59 69)(50 70 60 80)(81 140 91 130)(82 131 92 121)(83 122 93 132)(84 133 94 123)(85 124 95 134)(86 135 96 125)(87 126 97 136)(88 137 98 127)(89 128 99 138)(90 139 100 129)
(1 73 87 158)(2 74 88 159)(3 75 89 160)(4 76 90 141)(5 77 91 142)(6 78 92 143)(7 79 93 144)(8 80 94 145)(9 61 95 146)(10 62 96 147)(11 63 97 148)(12 64 98 149)(13 65 99 150)(14 66 100 151)(15 67 81 152)(16 68 82 153)(17 69 83 154)(18 70 84 155)(19 71 85 156)(20 72 86 157)(21 114 51 124)(22 115 52 125)(23 116 53 126)(24 117 54 127)(25 118 55 128)(26 119 56 129)(27 120 57 130)(28 101 58 131)(29 102 59 132)(30 103 60 133)(31 104 41 134)(32 105 42 135)(33 106 43 136)(34 107 44 137)(35 108 45 138)(36 109 46 139)(37 110 47 140)(38 111 48 121)(39 112 49 122)(40 113 50 123)
(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 10 97 86)(2 85 98 9)(3 8 99 84)(4 83 100 7)(5 6 81 82)(11 20 87 96)(12 95 88 19)(13 18 89 94)(14 93 90 17)(15 16 91 92)(21 34 41 54)(22 53 42 33)(23 32 43 52)(24 51 44 31)(25 30 45 50)(26 49 46 29)(27 28 47 48)(35 40 55 60)(36 59 56 39)(37 38 57 58)(61 74 156 149)(62 148 157 73)(63 72 158 147)(64 146 159 71)(65 70 160 145)(66 144 141 69)(67 68 142 143)(75 80 150 155)(76 154 151 79)(77 78 152 153)(101 140 121 120)(102 119 122 139)(103 138 123 118)(104 117 124 137)(105 136 125 116)(106 115 126 135)(107 134 127 114)(108 113 128 133)(109 132 129 112)(110 111 130 131)```

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M ··· 4R 4S 4T 4U 4V 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 1 1 1 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 C4○D4 D10 D10 D10 C4○D20 Q8×D5 D5×C4○D4 kernel C42.232D10 C4×Dic10 D5×C42 C42⋊D5 C20⋊Q8 Dic5.Q8 D10⋊Q8 D10⋊2Q8 Dic5⋊Q8 D10⋊3Q8 Q8×C20 C4×D5 C4×Q8 Dic5 C20 C42 C4⋊C4 C2×Q8 C4 C4 C2 # reps 1 3 1 2 1 2 2 1 1 1 1 4 2 4 4 6 6 2 16 4 4

Matrix representation of C42.232D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 28 32
,
 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 3 38 0 0 3 24 0 0 0 0 29 37 0 0 26 12
,
 24 3 0 0 40 17 0 0 0 0 12 4 0 0 36 29
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,28,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[3,3,0,0,38,24,0,0,0,0,29,26,0,0,37,12],[24,40,0,0,3,17,0,0,0,0,12,36,0,0,4,29] >;`

C42.232D10 in GAP, Magma, Sage, TeX

`C_4^2._{232}D_{10}`
`% in TeX`

`G:=Group("C4^2.232D10");`
`// GroupNames label`

`G:=SmallGroup(320,1250);`
`// by ID`

`G=gap.SmallGroup(320,1250);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,570,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^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^9>;`
`// generators/relations`

׿
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