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G = C4⋊D40order 320 = 26·5

The semidirect product of C4 and D40 acting via D40/D20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C4⋊D40
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4×D20 — C4⋊D40
 Lower central C5 — C10 — C2×C20 — C4⋊D40
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C4⋊D40
G = < a,b,c | a4=b40=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 854 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×11], C23 [×3], D5 [×4], C10 [×3], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×5], Dic5, C20 [×2], C20 [×2], C20, D10 [×10], C2×C10, D4⋊C4 [×2], C4⋊C8, C4×D4, C41D4, C2×D8 [×2], C40 [×2], C4×D5 [×2], D20 [×2], D20 [×9], C2×Dic5, C2×C20 [×3], C22×D5 [×3], C4⋊D8, D40 [×4], C4⋊Dic5, D10⋊C4, C4×C20, C2×C40 [×2], C2×C4×D5, C2×D20, C2×D20 [×2], C2×D20 [×2], D205C4 [×2], C5×C4⋊C8, C4×D20, C204D4, C2×D40 [×2], C4⋊D40
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C8⋊C22, D20 [×2], C22×D5, C4⋊D8, D40 [×2], C2×D20, D4×D5, Q82D5, C4⋊D20, C2×D40, C8⋊D10, C4⋊D40

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

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 C4○D4 D10 D10 D20 D40 C8⋊C22 D4×D5 Q8⋊2D5 C8⋊D10 kernel C4⋊D40 D20⋊5C4 C5×C4⋊C8 C4×D20 C20⋊4D4 C2×D40 D20 C2×C20 C4⋊C8 C20 C20 C42 C2×C8 C2×C4 C4 C10 C4 C4 C2 # reps 1 2 1 1 1 2 2 2 2 4 2 2 4 8 16 1 2 2 4

Matrix representation of C4⋊D40 in GL4(𝔽41) generated by

 0 32 0 0 32 0 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 40 0 0 0 0 0 8 35 0 0 11 38
,
 40 0 0 0 0 1 0 0 0 0 35 7 0 0 36 6
G:=sub<GL(4,GF(41))| [0,32,0,0,32,0,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,0,0,0,0,0,8,11,0,0,35,38],[40,0,0,0,0,1,0,0,0,0,35,36,0,0,7,6] >;

C4⋊D40 in GAP, Magma, Sage, TeX

C_4\rtimes D_{40}
% in TeX

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

G:=SmallGroup(320,470);
// by ID

G=gap.SmallGroup(320,470);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,226,1123,136,12550]);
// Polycyclic

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

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