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G = C409D4order 320 = 26·5

9th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C409D4, C52C810D4, C86(C5⋊D4), C55(C83D4), C408C46C2, C4.26(D4×D5), (C2×D40)⋊25C2, C20⋊D46C2, (C2×SD16)⋊2D5, (C2×C8).91D10, (C10×SD16)⋊4C2, (C2×D4).77D10, C20.179(C2×D4), (C2×Q8).58D10, C20.23D45C2, (C2×Dic5).82D4, C22.273(D4×D5), C2.23(C20⋊D4), C10.32(C41D4), C2.31(D40⋊C2), C10.81(C8⋊C22), (C2×C40).116C22, (C2×C20).453C23, (Q8×C10).82C22, (C2×D20).127C22, (D4×C10).102C22, (C4×Dic5).58C22, (C2×D4⋊D5)⋊21C2, (C2×Q8⋊D5)⋊19C2, C4.10(C2×C5⋊D4), (C2×C10).365(C2×D4), (C2×C4).542(C22×D5), (C2×C52C8).161C22, SmallGroup(320,803)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C409D4
C1C5C10C2×C10C2×C20C2×D20C2×D40 — C409D4
C5C10C2×C20 — C409D4
C1C22C2×C4C2×SD16

Generators and relations for C409D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a-1, cbc=b-1 >

Subgroups: 750 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, C4.4D4, C41D4, C2×D8, C2×SD16, C2×SD16, C52C8, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C83D4, D40, C2×C52C8, C4×Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C2×C40, C5×SD16, C2×D20, C2×C5⋊D4, D4×C10, Q8×C10, C408C4, C2×D40, C2×D4⋊D5, C20⋊D4, C2×Q8⋊D5, C20.23D4, C10×SD16, C409D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C41D4, C8⋊C22, C5⋊D4, C22×D5, C83D4, D4×D5, C2×C5⋊D4, D40⋊C2, C20⋊D4, C409D4

Smallest permutation representation of C409D4
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)
(1 102 146 73)(2 91 147 62)(3 120 148 51)(4 109 149 80)(5 98 150 69)(6 87 151 58)(7 116 152 47)(8 105 153 76)(9 94 154 65)(10 83 155 54)(11 112 156 43)(12 101 157 72)(13 90 158 61)(14 119 159 50)(15 108 160 79)(16 97 121 68)(17 86 122 57)(18 115 123 46)(19 104 124 75)(20 93 125 64)(21 82 126 53)(22 111 127 42)(23 100 128 71)(24 89 129 60)(25 118 130 49)(26 107 131 78)(27 96 132 67)(28 85 133 56)(29 114 134 45)(30 103 135 74)(31 92 136 63)(32 81 137 52)(33 110 138 41)(34 99 139 70)(35 88 140 59)(36 117 141 48)(37 106 142 77)(38 95 143 66)(39 84 144 55)(40 113 145 44)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(121 131)(122 130)(123 129)(124 128)(125 127)(132 160)(133 159)(134 158)(135 157)(136 156)(137 155)(138 154)(139 153)(140 152)(141 151)(142 150)(143 149)(144 148)(145 147)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444455888810···1010101010202020202020202040···40
size1111840402282020224420202···28888444488884···4

44 irreducible representations

dim11111111222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5D10D10D10C5⋊D4C8⋊C22D4×D5D4×D5D40⋊C2
kernelC409D4C408C4C2×D40C2×D4⋊D5C20⋊D4C2×Q8⋊D5C20.23D4C10×SD16C52C8C40C2×Dic5C2×SD16C2×C8C2×D4C2×Q8C8C10C4C22C2
# reps11111111222222282228

Matrix representation of C409D4 in GL8(𝔽41)

281925140000
221927270000
162713220000
141419220000
0000324324
0000374374
0000937324
0000437374
,
004000000
003510000
10000000
640000000
00001401516
000025273426
00002625140
00007152527
,
10000000
640000000
004000000
003510000
00001000
0000344000
000000400
00000071

G:=sub<GL(8,GF(41))| [28,22,16,14,0,0,0,0,19,19,27,14,0,0,0,0,25,27,13,19,0,0,0,0,14,27,22,22,0,0,0,0,0,0,0,0,32,37,9,4,0,0,0,0,4,4,37,37,0,0,0,0,32,37,32,37,0,0,0,0,4,4,4,4],[0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,14,25,26,7,0,0,0,0,0,27,25,15,0,0,0,0,15,34,14,25,0,0,0,0,16,26,0,27],[1,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,1] >;

C409D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_9D_4
% in TeX

G:=Group("C40:9D4");
// GroupNames label

G:=SmallGroup(320,803);
// by ID

G=gap.SmallGroup(320,803);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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