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## G = C42.100D10order 320 = 26·5

### 100th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.100D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C4×D20 — C42.100D10
 Lower central C5 — C2×C10 — C42.100D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.100D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c9 >

Subgroups: 1022 in 240 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C41D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C22.34C24, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, C20.6Q8, C4×D20, C204D4, D10⋊D4, D10.13D4, C207D4, C5×C42⋊C2, C42.100D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.34C24, C4○D20, C23×D5, C2×C4○D20, D48D10, C42.100D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20AB order 1 2 2 2 2 2 2 2 2 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 20 20 20 20 2 ··· 2 4 4 4 20 20 20 20 2 2 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 C4○D20 2+ 1+4 D4⋊8D10 kernel C42.100D10 C20.6Q8 C4×D20 C20⋊4D4 D10⋊D4 D10.13D4 C20⋊7D4 C5×C42⋊C2 C42⋊C2 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C4 C10 C2 # reps 1 1 2 1 4 4 2 1 2 4 4 4 4 2 16 2 8

Matrix representation of C42.100D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 32 0 0 0 0 37 39 0 0 0 0 0 0 2 32 0 0 0 0 37 39
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 22 0 3 0 0 0 0 22 0 3 0 0 3 0 19 0 0 0 0 3 0 19
,
 1 9 0 0 0 0 0 40 0 0 0 0 0 0 0 0 6 7 0 0 0 0 35 0 0 0 35 34 0 0 0 0 6 0 0 0
,
 40 32 0 0 0 0 23 1 0 0 0 0 0 0 0 0 16 30 0 0 0 0 12 25 0 0 25 11 0 0 0 0 29 16 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,37,0,0,0,0,32,39,0,0,0,0,0,0,2,37,0,0,0,0,32,39],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,22,0,3,0,0,0,0,22,0,3,0,0,3,0,19,0,0,0,0,3,0,19],[1,0,0,0,0,0,9,40,0,0,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,6,35,0,0,0,0,7,0,0,0],[40,23,0,0,0,0,32,1,0,0,0,0,0,0,0,0,25,29,0,0,0,0,11,16,0,0,16,12,0,0,0,0,30,25,0,0] >;`

C42.100D10 in GAP, Magma, Sage, TeX

`C_4^2._{100}D_{10}`
`% in TeX`

`G:=Group("C4^2.100D10");`
`// GroupNames label`

`G:=SmallGroup(320,1207);`
`// by ID`

`G=gap.SmallGroup(320,1207);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,297,136,12550]);`
`// 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,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;`
`// generators/relations`

׿
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