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## G = C42.153D10order 320 = 26·5

### 153rd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.153D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.153D10
 Lower central C5 — C2×C10 — C42.153D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.153D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 950 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×12], D4 [×10], C23 [×4], D5 [×5], C10 [×3], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×6], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×11], C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C422C2 [×2], C4×D5 [×8], D20 [×10], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], C22×D5 [×2], C22.47C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×C4×D5 [×2], C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×4], C4×D20 [×2], D5×C4⋊C4, C4⋊C47D5, D208C4 [×2], D10.13D4 [×2], C4⋊D20 [×2], C4⋊D20 [×2], C4⋊C4⋊D5 [×2], C5×C42.C2, C42.153D10
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], C22.47C24, Q82D5 [×2], C23×D5, C2×Q82D5, D5×C4○D4, D48D10, C42.153D10

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

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E ··· 4I 4J ··· 4O 4P 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 20 20 20 2 2 2 2 4 ··· 4 10 ··· 10 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 C4○D4 D10 D10 2+ 1+4 Q8⋊2D5 D5×C4○D4 D4⋊8D10 kernel C42.153D10 C4×D20 D5×C4⋊C4 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 C4⋊D20 C4⋊C4⋊D5 C5×C42.C2 C42.C2 C20 D10 C42 C4⋊C4 C10 C4 C2 C2 # reps 1 2 1 1 2 2 4 2 1 2 4 4 2 12 1 4 4 4

Matrix representation of C42.153D10 in GL6(𝔽41)

 40 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 0 9 0 0 0 0 9 0
,
 9 0 0 0 0 0 12 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 29 18 0 0 0 0 17 12 0 0 0 0 0 0 35 34 0 0 0 0 6 0 0 0 0 0 0 0 32 0 0 0 0 0 0 9
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 35 1 0 0 0 0 6 6 0 0 0 0 0 0 9 0 0 0 0 0 0 32

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0],[9,12,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[29,17,0,0,0,0,18,12,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;`

C42.153D10 in GAP, Magma, Sage, TeX

`C_4^2._{153}D_{10}`
`% in TeX`

`G:=Group("C4^2.153D10");`
`// GroupNames label`

`G:=SmallGroup(320,1367);`
`// by ID`

`G=gap.SmallGroup(320,1367);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,185,192,12550]);`
`// Polycyclic`

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

׿
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