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G = C4030D4order 320 = 26·5

2nd semidirect product of C40 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4030D4, C23.24D20, C56(C88D4), (C22×C8)⋊9D5, C814(C5⋊D4), C406C418C2, (C2×C10)⋊8SD16, (C2×C4).67D20, D205C42C2, (C22×C40)⋊13C2, C20.412(C2×D4), (C2×C20).355D4, (C2×C8).320D10, C207D4.5C2, C10.17(C4○D8), C20.44D42C2, C20.48D42C2, C221(C40⋊C2), C10.17(C2×SD16), C4.111(C4○D20), C20.227(C4○D4), C2.18(C207D4), C10.70(C4⋊D4), (C2×C40).393C22, (C2×C20).768C23, (C2×D20).20C22, (C22×C10).140D4, C22.131(C2×D20), (C22×C4).430D10, C4⋊Dic5.23C22, C2.17(D407C2), (C22×C20).518C22, (C2×Dic10).19C22, (C2×C40⋊C2)⋊21C2, C2.17(C2×C40⋊C2), C4.105(C2×C5⋊D4), (C2×C10).158(C2×D4), (C2×C4).716(C22×D5), SmallGroup(320,741)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4030D4
C1C5C10C2×C10C2×C20C2×D20C2×C40⋊C2 — C4030D4
C5C10C2×C20 — C4030D4
C1C22C22×C4C22×C8

Generators and relations for C4030D4
 G = < a,b,c | a40=b4=c2=1, bab-1=cac=a19, cbc=b-1 >

Subgroups: 550 in 124 conjugacy classes, 47 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C40 [×2], C40, Dic10 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C88D4, C40⋊C2 [×2], C10.D4, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C2×C40 [×2], C2×C40 [×2], C2×Dic10, C2×D20, C2×C5⋊D4, C22×C20, C20.44D4, C406C4, D205C4, C2×C40⋊C2, C20.48D4, C207D4, C22×C40, C4030D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, D20 [×2], C5⋊D4 [×2], C22×D5, C88D4, C40⋊C2 [×2], C2×D20, C4○D20, C2×C5⋊D4, C2×C40⋊C2, D407C2, C207D4, C4030D4

Smallest permutation representation of C4030D4
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)
(1 100 65 126)(2 119 66 145)(3 98 67 124)(4 117 68 143)(5 96 69 122)(6 115 70 141)(7 94 71 160)(8 113 72 139)(9 92 73 158)(10 111 74 137)(11 90 75 156)(12 109 76 135)(13 88 77 154)(14 107 78 133)(15 86 79 152)(16 105 80 131)(17 84 41 150)(18 103 42 129)(19 82 43 148)(20 101 44 127)(21 120 45 146)(22 99 46 125)(23 118 47 144)(24 97 48 123)(25 116 49 142)(26 95 50 121)(27 114 51 140)(28 93 52 159)(29 112 53 138)(30 91 54 157)(31 110 55 136)(32 89 56 155)(33 108 57 134)(34 87 58 153)(35 106 59 132)(36 85 60 151)(37 104 61 130)(38 83 62 149)(39 102 63 128)(40 81 64 147)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 49)(42 68)(43 47)(44 66)(46 64)(48 62)(50 60)(51 79)(52 58)(53 77)(54 56)(55 75)(57 73)(59 71)(61 69)(63 67)(70 80)(72 78)(74 76)(81 125)(82 144)(83 123)(84 142)(85 121)(86 140)(87 159)(88 138)(89 157)(90 136)(91 155)(92 134)(93 153)(94 132)(95 151)(96 130)(97 149)(98 128)(99 147)(100 126)(101 145)(102 124)(103 143)(104 122)(105 141)(106 160)(107 139)(108 158)(109 137)(110 156)(111 135)(112 154)(113 133)(114 152)(115 131)(116 150)(117 129)(118 148)(119 127)(120 146)

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

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

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

86 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A···8H10A···10N20A···20P40A···40AF
order12222224444444558···810···1020···2040···40
size111122402222404040222···22···22···22···2

86 irreducible representations

dim11111111222222222222222
type++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4SD16D10D10C4○D8C5⋊D4D20D20C4○D20C40⋊C2D407C2
kernelC4030D4C20.44D4C406C4D205C4C2×C40⋊C2C20.48D4C207D4C22×C40C40C2×C20C22×C10C22×C8C20C2×C10C2×C8C22×C4C10C8C2×C4C23C4C22C2
# reps1111111121122442484481616

Matrix representation of C4030D4 in GL4(𝔽41) generated by

161600
25200
001526
001515
,
202000
232100
00400
0001
,
63500
403500
0010
00040
G:=sub<GL(4,GF(41))| [16,25,0,0,16,2,0,0,0,0,15,15,0,0,26,15],[20,23,0,0,20,21,0,0,0,0,40,0,0,0,0,1],[6,40,0,0,35,35,0,0,0,0,1,0,0,0,0,40] >;

C4030D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{30}D_4
% in TeX

G:=Group("C40:30D4");
// GroupNames label

G:=SmallGroup(320,741);
// by ID

G=gap.SmallGroup(320,741);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=c*a*c=a^19,c*b*c=b^-1>;
// generators/relations

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