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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, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C40, C40, Dic10, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C88D4, C40⋊C2, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C40, C2×C40, 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, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, D20, C5⋊D4, C22×D5, C88D4, C40⋊C2, 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 42 87 139)(2 61 88 158)(3 80 89 137)(4 59 90 156)(5 78 91 135)(6 57 92 154)(7 76 93 133)(8 55 94 152)(9 74 95 131)(10 53 96 150)(11 72 97 129)(12 51 98 148)(13 70 99 127)(14 49 100 146)(15 68 101 125)(16 47 102 144)(17 66 103 123)(18 45 104 142)(19 64 105 121)(20 43 106 140)(21 62 107 159)(22 41 108 138)(23 60 109 157)(24 79 110 136)(25 58 111 155)(26 77 112 134)(27 56 113 153)(28 75 114 132)(29 54 115 151)(30 73 116 130)(31 52 117 149)(32 71 118 128)(33 50 119 147)(34 69 120 126)(35 48 81 145)(36 67 82 124)(37 46 83 143)(38 65 84 122)(39 44 85 141)(40 63 86 160)
(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)(41 160)(42 139)(43 158)(44 137)(45 156)(46 135)(47 154)(48 133)(49 152)(50 131)(51 150)(52 129)(53 148)(54 127)(55 146)(56 125)(57 144)(58 123)(59 142)(60 121)(61 140)(62 159)(63 138)(64 157)(65 136)(66 155)(67 134)(68 153)(69 132)(70 151)(71 130)(72 149)(73 128)(74 147)(75 126)(76 145)(77 124)(78 143)(79 122)(80 141)(81 93)(82 112)(83 91)(84 110)(85 89)(86 108)(88 106)(90 104)(92 102)(94 100)(95 119)(96 98)(97 117)(99 115)(101 113)(103 111)(105 109)(114 120)(116 118)

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

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

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

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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