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G = C4018D4order 320 = 26·5

18th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4018D4, Dic53M4(2), C89(C5⋊D4), C58(C86D4), (C8×Dic5)⋊31C2, C10.110(C4×D4), C20.443(C2×D4), (C2×C8).276D10, (C2×M4(2))⋊8D5, D101C839C2, C23.20(C4×D5), C10.57(C8○D4), (C10×M4(2))⋊7C2, C20.8Q841C2, C2.22(D5×M4(2)), C23.D5.20C4, D10⋊C4.27C4, C4.138(C4○D20), C20.254(C4○D4), C20.55D430C2, (C2×C20).868C23, (C2×C40).318C22, C10.D4.27C4, (C22×C4).137D10, C10.67(C2×M4(2)), C2.18(D20.2C4), (C22×C20).375C22, (C4×Dic5).317C22, (C2×C4).51(C4×D5), C2.25(C4×C5⋊D4), (C2×C8⋊D5)⋊25C2, (C2×C5⋊D4).20C4, (C4×C5⋊D4).17C2, C4.134(C2×C5⋊D4), C22.147(C2×C4×D5), (C2×C20).361(C2×C4), (C2×C4×D5).236C22, (C22×D5).30(C2×C4), (C2×C4).810(C22×D5), (C22×C10).136(C2×C4), (C2×C10).239(C22×C4), (C2×C52C8).332C22, (C2×Dic5).113(C2×C4), SmallGroup(320,755)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4018D4
C1C5C10C20C2×C20C2×C4×D5C4×C5⋊D4 — C4018D4
C5C2×C10 — C4018D4
C1C2×C4C2×M4(2)

Generators and relations for C4018D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a29, cbc=b-1 >

Subgroups: 382 in 122 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C2×M4(2), C52C8, C40, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C86D4, C8⋊D5, C2×C52C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C5×M4(2), C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D101C8, C20.55D4, C2×C8⋊D5, C4×C5⋊D4, C10×M4(2), C4018D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C5⋊D4, C22×D5, C86D4, C2×C4×D5, C4○D20, C2×C5⋊D4, D5×M4(2), D20.2C4, C4×C5⋊D4, C4018D4

Smallest permutation representation of C4018D4
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 71 121 98)(2 80 122 107)(3 49 123 116)(4 58 124 85)(5 67 125 94)(6 76 126 103)(7 45 127 112)(8 54 128 81)(9 63 129 90)(10 72 130 99)(11 41 131 108)(12 50 132 117)(13 59 133 86)(14 68 134 95)(15 77 135 104)(16 46 136 113)(17 55 137 82)(18 64 138 91)(19 73 139 100)(20 42 140 109)(21 51 141 118)(22 60 142 87)(23 69 143 96)(24 78 144 105)(25 47 145 114)(26 56 146 83)(27 65 147 92)(28 74 148 101)(29 43 149 110)(30 52 150 119)(31 61 151 88)(32 70 152 97)(33 79 153 106)(34 48 154 115)(35 57 155 84)(36 66 156 93)(37 75 157 102)(38 44 158 111)(39 53 159 120)(40 62 160 89)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 108)(42 97)(43 86)(44 115)(45 104)(46 93)(47 82)(48 111)(49 100)(50 89)(51 118)(52 107)(53 96)(54 85)(55 114)(56 103)(57 92)(58 81)(59 110)(60 99)(61 88)(62 117)(63 106)(64 95)(65 84)(66 113)(67 102)(68 91)(69 120)(70 109)(71 98)(72 87)(73 116)(74 105)(75 94)(76 83)(77 112)(78 101)(79 90)(80 119)(122 150)(123 139)(124 128)(125 157)(126 146)(127 135)(129 153)(130 142)(132 160)(133 149)(134 138)(136 156)(137 145)(140 152)(143 159)(144 148)(147 155)(154 158)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444445588888888888810···101010101020···202020202040···40
size1111420111141010101020222222441010101020202···244442···244444···4

68 irreducible representations

dim1111111111112222222222244
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D5M4(2)C4○D4D10D10C8○D4C5⋊D4C4×D5C4×D5C4○D20D5×M4(2)D20.2C4
kernelC4018D4C8×Dic5C20.8Q8D101C8C20.55D4C2×C8⋊D5C4×C5⋊D4C10×M4(2)C10.D4D10⋊C4C23.D5C2×C5⋊D4C40C2×M4(2)Dic5C20C2×C8C22×C4C10C8C2×C4C23C4C2C2
# reps1111111122222242424844844

Matrix representation of C4018D4 in GL4(𝔽41) generated by

403400
7700
0001
00320
,
174000
32400
00400
00040
,
1000
344000
0010
00040
G:=sub<GL(4,GF(41))| [40,7,0,0,34,7,0,0,0,0,0,32,0,0,1,0],[17,3,0,0,40,24,0,0,0,0,40,0,0,0,0,40],[1,34,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

C4018D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{18}D_4
% in TeX

G:=Group("C40:18D4");
// GroupNames label

G:=SmallGroup(320,755);
// by ID

G=gap.SmallGroup(320,755);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations

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