Copied to
clipboard

G = C402D4order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C402D4, C23.17D20, C56(C8⋊D4), C81(C5⋊D4), C405C418C2, (C2×C8).77D10, (C2×C4).51D20, C20.420(C2×D4), (C2×C20).297D4, (C2×M4(2))⋊1D5, D205C441C2, C207D4.17C2, (C10×M4(2))⋊1C2, (C2×C40).63C22, C20.230(C4○D4), C4.114(C4○D20), C20.44D441C2, C20.48D441C2, C2.22(C8⋊D10), C10.73(C4⋊D4), C2.21(C207D4), C10.22(C8⋊C22), (C2×C20).775C23, (C2×D20).22C22, (C22×C10).103D4, (C22×C4).142D10, C22.134(C2×D20), C4⋊Dic5.26C22, C2.22(C8.D10), C10.22(C8.C22), (C22×C20).304C22, (C2×Dic10).21C22, (C2×C40⋊C2)⋊2C2, C4.113(C2×C5⋊D4), (C2×C10).165(C2×D4), (C2×C4).724(C22×D5), SmallGroup(320,761)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C402D4
C1C5C10C2×C10C2×C20C2×D20C2×C40⋊C2 — C402D4
C5C10C2×C20 — C402D4
C1C22C22×C4C2×M4(2)

Generators and relations for C402D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a19, cbc=b-1 >

Subgroups: 550 in 120 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, D5, C10 [×3], C10, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), 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, C8⋊D4, C40⋊C2 [×2], C10.D4, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C2×C40 [×2], C5×M4(2) [×2], C2×Dic10, C2×D20, C2×C5⋊D4, C22×C20, C20.44D4, C405C4, D205C4, C2×C40⋊C2, C20.48D4, C207D4, C10×M4(2), C402D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, D20 [×2], C5⋊D4 [×2], C22×D5, C8⋊D4, C2×D20, C4○D20, C2×C5⋊D4, C8⋊D10, C8.D10, C207D4, C402D4

Smallest permutation representation of C402D4
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)
(1 142 116 68)(2 141 117 67)(3 140 118 66)(4 139 119 65)(5 138 120 64)(6 137 81 63)(7 136 82 62)(8 135 83 61)(9 134 84 60)(10 133 85 59)(11 132 86 58)(12 131 87 57)(13 130 88 56)(14 129 89 55)(15 128 90 54)(16 127 91 53)(17 126 92 52)(18 125 93 51)(19 124 94 50)(20 123 95 49)(21 122 96 48)(22 121 97 47)(23 160 98 46)(24 159 99 45)(25 158 100 44)(26 157 101 43)(27 156 102 42)(28 155 103 41)(29 154 104 80)(30 153 105 79)(31 152 106 78)(32 151 107 77)(33 150 108 76)(34 149 109 75)(35 148 110 74)(36 147 111 73)(37 146 112 72)(38 145 113 71)(39 144 114 70)(40 143 115 69)
(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)(41 149)(42 128)(43 147)(44 126)(45 145)(46 124)(47 143)(48 122)(49 141)(50 160)(51 139)(52 158)(53 137)(54 156)(55 135)(56 154)(57 133)(58 152)(59 131)(60 150)(61 129)(62 148)(63 127)(64 146)(65 125)(66 144)(67 123)(68 142)(69 121)(70 140)(71 159)(72 138)(73 157)(74 136)(75 155)(76 134)(77 153)(78 132)(79 151)(80 130)(81 91)(82 110)(83 89)(84 108)(85 87)(86 106)(88 104)(90 102)(92 100)(93 119)(94 98)(95 117)(97 115)(99 113)(101 111)(103 109)(105 107)(112 120)(114 118)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444455888810···101010101020···202020202040···40
size11114402244040402244442···244442···244444···4

56 irreducible representations

dim11111111222222222224444
type+++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10C5⋊D4D20D20C4○D20C8⋊C22C8.C22C8⋊D10C8.D10
kernelC402D4C20.44D4C405C4D205C4C2×C40⋊C2C20.48D4C207D4C10×M4(2)C40C2×C20C22×C10C2×M4(2)C20C2×C8C22×C4C8C2×C4C23C4C10C10C2C2
# reps11111111211224284481144

Matrix representation of C402D4 in GL6(𝔽41)

4000000
0400000
00462639
0034111610
001523735
002531730
,
16320000
24250000
0000341
0000347
0034100
0034700
,
100000
40400000
0034100
0034700
0000341
0000347

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,4,34,15,25,0,0,6,11,2,31,0,0,26,16,37,7,0,0,39,10,35,30],[16,24,0,0,0,0,32,25,0,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,34,34,0,0,0,0,1,7,0,0],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,1,7] >;

C402D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_2D_4
% in TeX

G:=Group("C40:2D4");
// GroupNames label

G:=SmallGroup(320,761);
// by ID

G=gap.SmallGroup(320,761);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,1684,102,12550]);
// Polycyclic

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

׿
×
𝔽