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## G = C42.97D10order 320 = 26·5

### 97th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.97D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C4⋊C4⋊7D5 — C42.97D10
 Lower central C5 — C2×C10 — C42.97D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.97D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 902 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×8], Q8 [×2], C23, C23 [×3], D5 [×3], C10 [×3], C10, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×6], C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×5], D10 [×9], C2×C10, C2×C10 [×3], C42⋊C2, C42⋊C2 [×3], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×6], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×D5 [×2], C22×C10, C22.49C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4, D10⋊C4 [×8], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, C42⋊D5 [×2], C4.D20 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], C20⋊Q8, C4⋊C47D5, D208C4, C4⋊D20, C4×C5⋊D4, C207D4, C5×C42⋊C2, C42.97D10
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], C22.49C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D48D10, C42.97D10

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

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 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 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 20 20 20 2 ··· 2 4 4 10 10 10 10 20 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⋊8D10 kernel C42.97D10 C42⋊D5 C4.D20 D10⋊D4 Dic5.5D4 C20⋊Q8 C4⋊C4⋊7D5 D20⋊8C4 C4⋊D20 C4×C5⋊D4 C20⋊7D4 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 C42.97D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 39 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 6 6 0 0 0 0 35 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 18 7 0 0 0 0 18 23
,
 35 35 0 0 0 0 40 6 0 0 0 0 0 0 0 32 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,39,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,18,0,0,0,0,7,23],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;`

C42.97D10 in GAP, Magma, Sage, TeX

`C_4^2._{97}D_{10}`
`% in TeX`

`G:=Group("C4^2.97D10");`
`// GroupNames label`

`G:=SmallGroup(320,1204);`
`// by ID`

`G=gap.SmallGroup(320,1204);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,136,12550]);`
`// 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,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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