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## G = C42.88D10order 320 = 26·5

### 88th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.88D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C42.88D10
 Lower central C5 — C2×C10 — C42.88D10
 Upper central C1 — C2×C4 — C42⋊C2

Generators and relations for C42.88D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 638 in 222 conjugacy classes, 115 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×14], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C42⋊C2, C42⋊C2, C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.37C23, C4×Dic5 [×2], C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×6], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C4×Dic10 [×4], Dic5.14D4 [×4], C20⋊Q8 [×2], C4.Dic10 [×2], C2×C4×Dic5, C23.21D10, C5×C42⋊C2, C42.88D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C22×Q8, C2×C4○D4 [×2], Dic10 [×4], C22×D5 [×7], C23.37C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D5×C4○D4 [×2], C42.88D10

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

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K ··· 4R 4S 4T 4U 4V 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 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + - + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 D10 D10 Dic10 D5×C4○D4 kernel C42.88D10 C4×Dic10 Dic5.14D4 C20⋊Q8 C4.Dic10 C2×C4×Dic5 C23.21D10 C5×C42⋊C2 C2×C20 C42⋊C2 Dic5 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C2 # reps 1 4 4 2 2 1 1 1 4 2 8 4 4 4 2 16 8

Matrix representation of C42.88D10 in GL4(𝔽41) generated by

 0 40 0 0 1 0 0 0 0 0 30 32 0 0 9 11
,
 32 0 0 0 0 32 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 40 0 0 0 0 7 7 0 0 34 40
,
 1 0 0 0 0 40 0 0 0 0 29 25 0 0 27 12
`G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,30,9,0,0,32,11],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,7,34,0,0,7,40],[1,0,0,0,0,40,0,0,0,0,29,27,0,0,25,12] >;`

C42.88D10 in GAP, Magma, Sage, TeX

`C_4^2._{88}D_{10}`
`% in TeX`

`G:=Group("C4^2.88D10");`
`// GroupNames label`

`G:=SmallGroup(320,1189);`
`// by ID`

`G=gap.SmallGroup(320,1189);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,675,570,80,12550]);`
`// Polycyclic`

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

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