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G = D40.3C4order 320 = 26·5

1st non-split extension by D40 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D40.3C4
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — D40⋊7C2 — D40.3C4
 Lower central C5 — C10 — C20 — C40 — D40.3C4
 Upper central C1 — C4 — C2×C4 — C2×C8 — C2×C16

Generators and relations for D40.3C4
G = < a,b,c | a40=b2=1, c4=a10, bab=a-1, ac=ca, cbc-1=a15b >

Smallest permutation representation of D40.3C4
On 160 points
Generators in S160
```(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 83)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 160)(137 159)(138 158)(139 157)(140 156)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)
(1 90 49 151 11 100 59 121 21 110 69 131 31 120 79 141)(2 91 50 152 12 101 60 122 22 111 70 132 32 81 80 142)(3 92 51 153 13 102 61 123 23 112 71 133 33 82 41 143)(4 93 52 154 14 103 62 124 24 113 72 134 34 83 42 144)(5 94 53 155 15 104 63 125 25 114 73 135 35 84 43 145)(6 95 54 156 16 105 64 126 26 115 74 136 36 85 44 146)(7 96 55 157 17 106 65 127 27 116 75 137 37 86 45 147)(8 97 56 158 18 107 66 128 28 117 76 138 38 87 46 148)(9 98 57 159 19 108 67 129 29 118 77 139 39 88 47 149)(10 99 58 160 20 109 68 130 30 119 78 140 40 89 48 150)```

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

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

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

86 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 10A ··· 10F 16A ··· 16H 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 2 40 1 1 2 40 2 2 2 2 2 2 40 40 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

86 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D5 D8 SD16 D10 C4×D5 C5⋊D4 D20 D8.C4 D40 C40⋊C2 D40.3C4 kernel D40.3C4 C40.6C4 C2×C80 D40⋊7C2 D40 Dic20 C40 C2×C20 C2×C16 C20 C2×C10 C2×C8 C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 2 2 2 2 4 4 4 8 8 8 32

Matrix representation of D40.3C4 in GL2(𝔽241) generated by

 20 194 47 227
,
 14 42 47 227
,
 101 201 40 190
`G:=sub<GL(2,GF(241))| [20,47,194,227],[14,47,42,227],[101,40,201,190] >;`

D40.3C4 in GAP, Magma, Sage, TeX

`D_{40}._3C_4`
`% in TeX`

`G:=Group("D40.3C4");`
`// GroupNames label`

`G:=SmallGroup(320,68);`
`// by ID`

`G=gap.SmallGroup(320,68);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,268,1123,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^2=1,c^4=a^10,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;`
`// generators/relations`

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