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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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C40, C40, Dic10, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊D4, C40⋊C2, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C40, C5×M4(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, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, D20, C5⋊D4, 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 56 130 100)(2 55 131 99)(3 54 132 98)(4 53 133 97)(5 52 134 96)(6 51 135 95)(7 50 136 94)(8 49 137 93)(9 48 138 92)(10 47 139 91)(11 46 140 90)(12 45 141 89)(13 44 142 88)(14 43 143 87)(15 42 144 86)(16 41 145 85)(17 80 146 84)(18 79 147 83)(19 78 148 82)(20 77 149 81)(21 76 150 120)(22 75 151 119)(23 74 152 118)(24 73 153 117)(25 72 154 116)(26 71 155 115)(27 70 156 114)(28 69 157 113)(29 68 158 112)(30 67 159 111)(31 66 160 110)(32 65 121 109)(33 64 122 108)(34 63 123 107)(35 62 124 106)(36 61 125 105)(37 60 126 104)(38 59 127 103)(39 58 128 102)(40 57 129 101)
(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 95)(42 114)(43 93)(44 112)(45 91)(46 110)(47 89)(48 108)(49 87)(50 106)(51 85)(52 104)(53 83)(54 102)(55 81)(56 100)(57 119)(58 98)(59 117)(60 96)(61 115)(62 94)(63 113)(64 92)(65 111)(66 90)(67 109)(68 88)(69 107)(70 86)(71 105)(72 84)(73 103)(74 82)(75 101)(76 120)(77 99)(78 118)(79 97)(80 116)(121 159)(122 138)(123 157)(124 136)(125 155)(126 134)(127 153)(128 132)(129 151)(131 149)(133 147)(135 145)(137 143)(139 141)(140 160)(142 158)(144 156)(146 154)(148 152)

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

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

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

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

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