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G = C42.177D10order 320 = 26·5

177th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.177D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊2Q8 — C42.177D10
 Lower central C5 — C2×C10 — C42.177D10
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.177D10
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c9 >

Subgroups: 678 in 212 conjugacy classes, 99 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×4], C20 [×5], D10 [×6], C2×C10, C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5 [×2], C22.50C24, C4×Dic5 [×6], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, Q8×C10 [×2], C4×Dic10, C4×D20, C4⋊C47D5 [×2], D102Q8 [×2], C4⋊C4⋊D5 [×4], Q8×Dic5 [×2], C20.23D4 [×2], C5×C4⋊Q8, C42.177D10
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, D42D5 [×2], Q82D5 [×2], C23×D5, C2×D42D5, C2×Q82D5, Q8.10D10, C42.177D10

Smallest permutation representation of C42.177D10
On 160 points
Generators in S160
```(1 134 89 119)(2 120 90 135)(3 136 91 101)(4 102 92 137)(5 138 93 103)(6 104 94 139)(7 140 95 105)(8 106 96 121)(9 122 97 107)(10 108 98 123)(11 124 99 109)(12 110 100 125)(13 126 81 111)(14 112 82 127)(15 128 83 113)(16 114 84 129)(17 130 85 115)(18 116 86 131)(19 132 87 117)(20 118 88 133)(21 160 62 46)(22 47 63 141)(23 142 64 48)(24 49 65 143)(25 144 66 50)(26 51 67 145)(27 146 68 52)(28 53 69 147)(29 148 70 54)(30 55 71 149)(31 150 72 56)(32 57 73 151)(33 152 74 58)(34 59 75 153)(35 154 76 60)(36 41 77 155)(37 156 78 42)(38 43 79 157)(39 158 80 44)(40 45 61 159)
(1 58 99 142)(2 143 100 59)(3 60 81 144)(4 145 82 41)(5 42 83 146)(6 147 84 43)(7 44 85 148)(8 149 86 45)(9 46 87 150)(10 151 88 47)(11 48 89 152)(12 153 90 49)(13 50 91 154)(14 155 92 51)(15 52 93 156)(16 157 94 53)(17 54 95 158)(18 159 96 55)(19 56 97 160)(20 141 98 57)(21 117 72 122)(22 123 73 118)(23 119 74 124)(24 125 75 120)(25 101 76 126)(26 127 77 102)(27 103 78 128)(28 129 79 104)(29 105 80 130)(30 131 61 106)(31 107 62 132)(32 133 63 108)(33 109 64 134)(34 135 65 110)(35 111 66 136)(36 137 67 112)(37 113 68 138)(38 139 69 114)(39 115 70 140)(40 121 71 116)
(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 89 98)(2 97 90 9)(3 8 91 96)(4 95 92 7)(5 6 93 94)(11 20 99 88)(12 87 100 19)(13 18 81 86)(14 85 82 17)(15 16 83 84)(21 24 62 65)(22 64 63 23)(25 40 66 61)(26 80 67 39)(27 38 68 79)(28 78 69 37)(29 36 70 77)(30 76 71 35)(31 34 72 75)(32 74 73 33)(41 148 155 54)(42 53 156 147)(43 146 157 52)(44 51 158 145)(45 144 159 50)(46 49 160 143)(47 142 141 48)(55 154 149 60)(56 59 150 153)(57 152 151 58)(101 106 136 121)(102 140 137 105)(103 104 138 139)(107 120 122 135)(108 134 123 119)(109 118 124 133)(110 132 125 117)(111 116 126 131)(112 130 127 115)(113 114 128 129)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4I 4J ··· 4Q 4R 4S 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 20 20 2 2 2 2 4 ··· 4 10 ··· 10 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2- 1+4 D4⋊2D5 Q8⋊2D5 Q8.10D10 kernel C42.177D10 C4×Dic10 C4×D20 C4⋊C4⋊7D5 D10⋊2Q8 C4⋊C4⋊D5 Q8×Dic5 C20.23D4 C5×C4⋊Q8 C4⋊Q8 C20 C42 C4⋊C4 C2×Q8 C10 C4 C4 C2 # reps 1 1 1 2 2 4 2 2 1 2 8 2 8 4 1 4 4 4

Matrix representation of C42.177D10 in GL6(𝔽41)

 32 0 0 0 0 0 6 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 0 0 0 0 0 35 32 0 0 0 0 0 0 21 9 0 0 0 0 1 20 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 26 37 0 0 0 0 15 15 0 0 0 0 0 0 25 40 0 0 0 0 11 16 0 0 0 0 0 0 40 7 0 0 0 0 34 7
,
 15 4 0 0 0 0 5 26 0 0 0 0 0 0 16 1 0 0 0 0 32 25 0 0 0 0 0 0 40 0 0 0 0 0 34 1

`G:=sub<GL(6,GF(41))| [32,6,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,35,0,0,0,0,0,32,0,0,0,0,0,0,21,1,0,0,0,0,9,20,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[26,15,0,0,0,0,37,15,0,0,0,0,0,0,25,11,0,0,0,0,40,16,0,0,0,0,0,0,40,34,0,0,0,0,7,7],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,16,32,0,0,0,0,1,25,0,0,0,0,0,0,40,34,0,0,0,0,0,1] >;`

C42.177D10 in GAP, Magma, Sage, TeX

`C_4^2._{177}D_{10}`
`% in TeX`

`G:=Group("C4^2.177D10");`
`// GroupNames label`

`G:=SmallGroup(320,1404);`
`// by ID`

`G=gap.SmallGroup(320,1404);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,1571,297,136,12550]);`
`// Polycyclic`

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

׿
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