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## G = C42.98D10order 320 = 26·5

### 98th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.98D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C4⋊C4⋊7D5 — C42.98D10
 Lower central C5 — C2×C10 — C42.98D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.98D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 662 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, D5, C10 [×3], C10, C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4, C22×C4, C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×5], D10 [×3], C2×C10, C2×C10 [×3], C42⋊C2, C42⋊C2, C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, Dic10 [×6], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×C10, C22.50C24, C4×Dic5, C4×Dic5 [×4], C10.D4, C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×4], C23.D5, C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C4×Dic10 [×2], C422D5 [×2], C23.D10 [×2], Dic5.5D4 [×2], Dic53Q8, C20⋊Q8, C4⋊C47D5, D102Q8, C20.48D4, C4×C5⋊D4, C5×C42⋊C2, C42.98D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.50C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.98D10

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 C4○D20 2- 1+4 D5×C4○D4 D4.10D10 kernel C42.98D10 C4×Dic10 C42⋊2D5 C23.D10 Dic5.5D4 Dic5⋊3Q8 C20⋊Q8 C4⋊C4⋊7D5 D10⋊2Q8 C20.48D4 C4×C5⋊D4 C5×C42⋊C2 C42⋊C2 Dic5 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C4 C10 C2 C2 # reps 1 2 2 2 2 1 1 1 1 1 1 1 2 4 4 4 4 4 2 16 1 4 4

Matrix representation of C42.98D10 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 32 32
,
 24 40 0 0 1 17 0 0 0 0 9 0 0 0 0 9
,
 1 34 0 0 7 34 0 0 0 0 9 18 0 0 0 32
,
 34 1 0 0 34 7 0 0 0 0 9 0 0 0 0 9
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,32,0,0,0,32],[24,1,0,0,40,17,0,0,0,0,9,0,0,0,0,9],[1,7,0,0,34,34,0,0,0,0,9,0,0,0,18,32],[34,34,0,0,1,7,0,0,0,0,9,0,0,0,0,9] >;`

C42.98D10 in GAP, Magma, Sage, TeX

`C_4^2._{98}D_{10}`
`% in TeX`

`G:=Group("C4^2.98D10");`
`// GroupNames label`

`G:=SmallGroup(320,1205);`
`// by ID`

`G=gap.SmallGroup(320,1205);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,1571,136,12550]);`
`// Polycyclic`

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

׿
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