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## G = C42.93D10order 320 = 26·5

### 93rd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.93D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.93D10
 Lower central C5 — C2×C10 — C42.93D10
 Upper central C1 — C2×C4 — C42⋊C2

Generators and relations for C42.93D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=b-1, dcd-1=c9 >

Subgroups: 782 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×3], C10 [×3], C10, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×5], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×2], C22×D5 [×2], C22×C10, C23.36C23, C4×Dic5 [×4], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×4], C2×D20, C2×C5⋊D4 [×2], C22×C20, C4×Dic10, D5×C42, C42⋊D5, C4×D20, C23.D10, D10.12D4, D10⋊D4, Dic5.5D4, Dic5.Q8, D10.13D4, D10⋊Q8, C4⋊C4⋊D5, C4×C5⋊D4 [×2], C5×C42⋊C2, C42.93D10
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], C23×D5, C2×C4○D20, D5×C4○D4 [×2], C42.93D10

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

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L ··· 4Q 4R 4S 4T 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20AB order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 10 10 20 1 1 1 1 2 2 2 2 4 4 4 10 ··· 10 20 20 20 2 2 2 ··· 2 4 4 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 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 C4○D20 D5×C4○D4 kernel C42.93D10 C4×Dic10 D5×C42 C42⋊D5 C4×D20 C23.D10 D10.12D4 D10⋊D4 Dic5.5D4 Dic5.Q8 D10.13D4 D10⋊Q8 C4⋊C4⋊D5 C4×C5⋊D4 C5×C42⋊C2 C42⋊C2 Dic5 C20 D10 C42 C22⋊C4 C4⋊C4 C22×C4 C4 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 4 4 4 4 4 4 2 16 8

Matrix representation of C42.93D10 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 40 0 0 0 0 40
,
 32 2 0 0 1 9 0 0 0 0 11 32 0 0 9 30
,
 9 39 0 0 0 32 0 0 0 0 9 19 0 0 22 19
,
 9 0 0 0 0 9 0 0 0 0 24 40 0 0 3 17
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[32,1,0,0,2,9,0,0,0,0,11,9,0,0,32,30],[9,0,0,0,39,32,0,0,0,0,9,22,0,0,19,19],[9,0,0,0,0,9,0,0,0,0,24,3,0,0,40,17] >;`

C42.93D10 in GAP, Magma, Sage, TeX

`C_4^2._{93}D_{10}`
`% in TeX`

`G:=Group("C4^2.93D10");`
`// GroupNames label`

`G:=SmallGroup(320,1200);`
`// by ID`

`G=gap.SmallGroup(320,1200);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,297,12550]);`
`// Polycyclic`

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

׿
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