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## G = C42.144D10order 320 = 26·5

### 144th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.144D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C42⋊D5 — C42.144D10
 Lower central C5 — C2×C10 — C42.144D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.144D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 734 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, D5, C10 [×3], C10 [×2], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], Dic5 [×7], C20 [×2], C20 [×4], D10 [×3], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C4.4D4 [×2], C422C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10 [×2], C22.36C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C202Q8, C42⋊D5, Dic5.14D4 [×2], C23.D10 [×2], Dic5.5D4 [×2], C22.D20 [×2], D4×Dic5, C202D4, Q8×Dic5, D103Q8, C5×C4.4D4, C42.144D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.36C24, D42D5 [×2], C23×D5, C2×D42D5, D48D10, D4.10D10, C42.144D10

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊2D5 D4⋊8D10 D4.10D10 kernel C42.144D10 C20⋊2Q8 C42⋊D5 Dic5.14D4 C23.D10 Dic5.5D4 C22.D20 D4×Dic5 C20⋊2D4 Q8×Dic5 D10⋊3Q8 C5×C4.4D4 C4.4D4 C20 C42 C22⋊C4 C2×D4 C2×Q8 C10 C10 C4 C2 C2 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 4 2 8 2 2 1 1 4 4 4

Matrix representation of C42.144D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 26 0 29 0 0 0 0 26 0 29 0 0 12 0 15 0 0 0 0 12 0 15
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 30 12 24 26 0 0 19 11 7 17 0 0 17 15 11 29 0 0 34 24 22 30
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 35 6 0 0 0 7 0 0 0 0 35 6 0 0
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 35 7 0 0 0 0 36 6 0 0 35 7 0 0 0 0 36 6 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,26,0,12,0,0,0,0,26,0,12,0,0,29,0,15,0,0,0,0,29,0,15],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,30,19,17,34,0,0,12,11,15,24,0,0,24,7,11,22,0,0,26,17,29,30],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,35,0,0,0,0,7,6,0,0,0,35,0,0,0,0,7,6,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,36,0,0,0,0,7,6,0,0,35,36,0,0,0,0,7,6,0,0] >;`

C42.144D10 in GAP, Magma, Sage, TeX

`C_4^2._{144}D_{10}`
`% in TeX`

`G:=Group("C4^2.144D10");`
`// GroupNames label`

`G:=SmallGroup(320,1354);`
`// by ID`

`G=gap.SmallGroup(320,1354);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,409,12550]);`
`// 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,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;`
`// generators/relations`

׿
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