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## G = C42.99D10order 320 = 26·5

### 99th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.99D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C4×D20 — C42.99D10
 Lower central C5 — C2×C10 — C42.99D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.99D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 782 in 216 conjugacy classes, 95 normal (51 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], D5 [×2], C10 [×3], C10, C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8 [×3], Dic5 [×6], C20 [×2], C20 [×5], D10 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C422C2 [×2], C4⋊Q8, Dic10 [×4], C4×D5 [×2], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×2], C2×C20 [×6], C2×C20 [×2], C22×D5 [×2], C22×C10, C22.36C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, C4×Dic10, C202Q8, C4×D20, C4.D20, D10.12D4 [×2], Dic5.5D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], C20.48D4, C207D4, C5×C42⋊C2, C42.99D10
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, C4○D20 [×2], C23×D5, C2×C4○D20, D48D10, D4.10D10, C42.99D10

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 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 C4○D20 2+ 1+4 2- 1+4 D4⋊8D10 D4.10D10 kernel C42.99D10 C4×Dic10 C20⋊2Q8 C4×D20 C4.D20 D10.12D4 Dic5.5D4 D10⋊Q8 C4⋊C4⋊D5 C20.48D4 C20⋊7D4 C5×C42⋊C2 C42⋊C2 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C4 C10 C10 C2 C2 # reps 1 1 1 1 1 2 2 2 2 1 1 1 2 4 4 4 4 2 16 1 1 4 4

Matrix representation of C42.99D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 30 9 0 0 0 0 32 11 0 0 0 0 0 0 30 9 0 0 0 0 32 11
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 14 0 5 22 0 0 0 14 19 36 0 0 36 19 27 0 0 0 22 5 0 27
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 40 7 0 0 0 0 34 7 0 0
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 34 1 0 0 1 0 0 0 0 0 7 40 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,32,0,0,0,0,9,11,0,0,0,0,0,0,30,32,0,0,0,0,9,11],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,14,0,36,22,0,0,0,14,19,5,0,0,5,19,27,0,0,0,22,36,0,27],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,40,34,0,0,0,0,7,7,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,40,0,0,40,34,0,0,0,0,0,1,0,0] >;`

C42.99D10 in GAP, Magma, Sage, TeX

`C_4^2._{99}D_{10}`
`% in TeX`

`G:=Group("C4^2.99D10");`
`// GroupNames label`

`G:=SmallGroup(320,1206);`
`// by ID`

`G=gap.SmallGroup(320,1206);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,297,12550]);`
`// Polycyclic`

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

׿
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