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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 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2), C2×M4(2), C52C8 [×2], C40 [×2], C40, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C86D4, C8⋊D5 [×2], C2×C52C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C5×M4(2) [×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 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C5⋊D4 [×2], 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 84 48 122)(2 93 49 131)(3 102 50 140)(4 111 51 149)(5 120 52 158)(6 89 53 127)(7 98 54 136)(8 107 55 145)(9 116 56 154)(10 85 57 123)(11 94 58 132)(12 103 59 141)(13 112 60 150)(14 81 61 159)(15 90 62 128)(16 99 63 137)(17 108 64 146)(18 117 65 155)(19 86 66 124)(20 95 67 133)(21 104 68 142)(22 113 69 151)(23 82 70 160)(24 91 71 129)(25 100 72 138)(26 109 73 147)(27 118 74 156)(28 87 75 125)(29 96 76 134)(30 105 77 143)(31 114 78 152)(32 83 79 121)(33 92 80 130)(34 101 41 139)(35 110 42 148)(36 119 43 157)(37 88 44 126)(38 97 45 135)(39 106 46 144)(40 115 47 153)
(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 45)(42 74)(43 63)(44 52)(46 70)(47 59)(49 77)(50 66)(51 55)(53 73)(54 62)(56 80)(57 69)(60 76)(61 65)(64 72)(67 79)(71 75)(81 155)(82 144)(83 133)(84 122)(85 151)(86 140)(87 129)(88 158)(89 147)(90 136)(91 125)(92 154)(93 143)(94 132)(95 121)(96 150)(97 139)(98 128)(99 157)(100 146)(101 135)(102 124)(103 153)(104 142)(105 131)(106 160)(107 149)(108 138)(109 127)(110 156)(111 145)(112 134)(113 123)(114 152)(115 141)(116 130)(117 159)(118 148)(119 137)(120 126)

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

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

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

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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