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## G = C42.136D10order 320 = 26·5

### 136th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.136D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.13D4 — C42.136D10
 Lower central C5 — C2×C10 — C42.136D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.136D10
G = < a,b,c,d | a4=b4=d2=1, c10=a2, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=b-1, dcd=a2c9 >

Subgroups: 934 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4×D4, C4×Q8, C4×Q8, C22.D4, C4.4D4, C41D4, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.53C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C4×Dic10, C4×D20, C204D4, C4.D20, D208C4, D10.13D4, C20.23D4, Q8×C20, C42.136D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.53C24, C4○D20, Q82D5, C23×D5, C2×C4○D20, C2×Q82D5, D48D10, C42.136D10

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

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

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

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

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 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 2 2 2 4 ··· 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 20 20 2 ··· 2 4 4 4 10 10 10 10 20 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 C4○D20 2+ 1+4 Q8⋊2D5 D4⋊8D10 kernel C42.136D10 C4×Dic10 C4×D20 C20⋊4D4 C4.D20 D20⋊8C4 D10.13D4 C20.23D4 Q8×C20 C4×Q8 C20 C42 C4⋊C4 C2×Q8 C4 C10 C4 C2 # reps 1 1 2 1 2 2 4 2 1 2 8 6 6 2 16 1 4 4

Matrix representation of C42.136D10 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 9 23 0 0 0 32
,
 2 4 0 0 9 39 0 0 0 0 40 0 0 0 0 40
,
 25 12 0 0 27 13 0 0 0 0 1 39 0 0 1 40
,
 6 5 0 0 34 35 0 0 0 0 1 0 0 0 1 40
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,23,32],[2,9,0,0,4,39,0,0,0,0,40,0,0,0,0,40],[25,27,0,0,12,13,0,0,0,0,1,1,0,0,39,40],[6,34,0,0,5,35,0,0,0,0,1,1,0,0,0,40] >;`

C42.136D10 in GAP, Magma, Sage, TeX

`C_4^2._{136}D_{10}`
`% in TeX`

`G:=Group("C4^2.136D10");`
`// GroupNames label`

`G:=SmallGroup(320,1257);`
`// by ID`

`G=gap.SmallGroup(320,1257);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,184,1571,192,12550]);`
`// Polycyclic`

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

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