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G = C40.36D4order 320 = 26·5

36th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.36D4, (C2×Q16)⋊8D5, (C10×Q16)⋊9C2, C406C422C2, (C2×C8).95D10, C55(C8.D4), C8.5(C5⋊D4), C20.187(C2×D4), (C2×Q8).65D10, Q8⋊Dic536C2, (C2×Dic5).85D4, (C22×D5).49D4, C22.280(D4×D5), C20.108(C4○D4), C4.37(D42D5), C2.23(C202D4), (C2×C20).463C23, (C2×C40).150C22, D103Q8.10C2, (Q8×C10).92C22, C10.122(C4⋊D4), C2.30(Q16⋊D5), C10.80(C8.C22), C4⋊Dic5.186C22, C4.86(C2×C5⋊D4), (C2×C8⋊D5).5C2, (C2×C4×D5).58C22, (C2×C10).374(C2×D4), (C2×C4).551(C22×D5), (C2×C52C8).167C22, SmallGroup(320,816)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.36D4
C1C5C10C2×C10C2×C20C2×C4×D5C2×C8⋊D5 — C40.36D4
C5C10C2×C20 — C40.36D4
C1C22C2×C4C2×Q16

Generators and relations for C40.36D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a29, cbc=a20b-1 >

Subgroups: 406 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], Q8 [×4], C23, D5, C10, C10 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8, M4(2) [×2], Q16 [×2], C22×C4, C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, Q8⋊C4 [×2], C4.Q8, C22⋊Q8 [×2], C2×M4(2), C2×Q16, C52C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C8.D4, C8⋊D5 [×2], C2×C52C8, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C2×C40, C5×Q16 [×2], C2×C4×D5, Q8×C10 [×2], C406C4, Q8⋊Dic5 [×2], C2×C8⋊D5, D103Q8 [×2], C10×Q16, C40.36D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8.C22 [×2], C5⋊D4 [×2], C22×D5, C8.D4, D4×D5, D42D5, C2×C5⋊D4, Q16⋊D5 [×2], C202D4, C40.36D4

Smallest permutation representation of C40.36D4
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)
(1 154 67 91)(2 133 68 110)(3 152 69 89)(4 131 70 108)(5 150 71 87)(6 129 72 106)(7 148 73 85)(8 127 74 104)(9 146 75 83)(10 125 76 102)(11 144 77 81)(12 123 78 100)(13 142 79 119)(14 121 80 98)(15 140 41 117)(16 159 42 96)(17 138 43 115)(18 157 44 94)(19 136 45 113)(20 155 46 92)(21 134 47 111)(22 153 48 90)(23 132 49 109)(24 151 50 88)(25 130 51 107)(26 149 52 86)(27 128 53 105)(28 147 54 84)(29 126 55 103)(30 145 56 82)(31 124 57 101)(32 143 58 120)(33 122 59 99)(34 141 60 118)(35 160 61 97)(36 139 62 116)(37 158 63 95)(38 137 64 114)(39 156 65 93)(40 135 66 112)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 73)(42 62)(43 51)(44 80)(45 69)(46 58)(48 76)(49 65)(50 54)(52 72)(53 61)(55 79)(56 68)(59 75)(60 64)(63 71)(66 78)(70 74)(81 124)(82 153)(83 142)(84 131)(85 160)(86 149)(87 138)(88 127)(89 156)(90 145)(91 134)(92 123)(93 152)(94 141)(95 130)(96 159)(97 148)(98 137)(99 126)(100 155)(101 144)(102 133)(103 122)(104 151)(105 140)(106 129)(107 158)(108 147)(109 136)(110 125)(111 154)(112 143)(113 132)(114 121)(115 150)(116 139)(117 128)(118 157)(119 146)(120 135)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444455888810···102020202020···2040···40
size1111202288204040224420202···244448···84···4

44 irreducible representations

dim111111222222224444
type++++++++++++--+
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10C5⋊D4C8.C22D42D5D4×D5Q16⋊D5
kernelC40.36D4C406C4Q8⋊Dic5C2×C8⋊D5D103Q8C10×Q16C40C2×Dic5C22×D5C2×Q16C20C2×C8C2×Q8C8C10C4C22C2
# reps112121211222482228

Matrix representation of C40.36D4 in GL8(𝔽41)

1203500000
103835350000
21020210000
182114230000
0000105105
000025352535
00003136105
00001662535
,
6403610000
134760000
3821100000
3122100000
00007402420
000040342017
00002420341
0000201717
,
341000000
347000000
3204000000
9193510000
00006700
0000363500
00000067
0000003635

G:=sub<GL(8,GF(41))| [1,10,21,18,0,0,0,0,20,38,0,21,0,0,0,0,35,35,20,14,0,0,0,0,0,35,21,23,0,0,0,0,0,0,0,0,10,25,31,16,0,0,0,0,5,35,36,6,0,0,0,0,10,25,10,25,0,0,0,0,5,35,5,35],[6,1,38,31,0,0,0,0,40,34,21,22,0,0,0,0,36,7,1,1,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,7,40,24,20,0,0,0,0,40,34,20,17,0,0,0,0,24,20,34,1,0,0,0,0,20,17,1,7],[34,34,3,9,0,0,0,0,1,7,20,19,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,36,0,0,0,0,0,0,7,35,0,0,0,0,0,0,0,0,6,36,0,0,0,0,0,0,7,35] >;

C40.36D4 in GAP, Magma, Sage, TeX

C_{40}._{36}D_4
% in TeX

G:=Group("C40.36D4");
// GroupNames label

G:=SmallGroup(320,816);
// by ID

G=gap.SmallGroup(320,816);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,219,184,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^29,c*b*c=a^20*b^-1>;
// generators/relations

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