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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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C8.D4, C8⋊D5, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C2×C40, C5×Q16, C2×C4×D5, Q8×C10, C406C4, Q8⋊Dic5, C2×C8⋊D5, D103Q8, C10×Q16, C40.36D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8.C22, C5⋊D4, C22×D5, C8.D4, D4×D5, D42D5, C2×C5⋊D4, Q16⋊D5, 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)
(1 126 47 88)(2 145 48 107)(3 124 49 86)(4 143 50 105)(5 122 51 84)(6 141 52 103)(7 160 53 82)(8 139 54 101)(9 158 55 120)(10 137 56 99)(11 156 57 118)(12 135 58 97)(13 154 59 116)(14 133 60 95)(15 152 61 114)(16 131 62 93)(17 150 63 112)(18 129 64 91)(19 148 65 110)(20 127 66 89)(21 146 67 108)(22 125 68 87)(23 144 69 106)(24 123 70 85)(25 142 71 104)(26 121 72 83)(27 140 73 102)(28 159 74 81)(29 138 75 100)(30 157 76 119)(31 136 77 98)(32 155 78 117)(33 134 79 96)(34 153 80 115)(35 132 41 94)(36 151 42 113)(37 130 43 92)(38 149 44 111)(39 128 45 90)(40 147 46 109)
(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)(81 143)(82 132)(83 121)(84 150)(85 139)(86 128)(87 157)(88 146)(89 135)(90 124)(91 153)(92 142)(93 131)(94 160)(95 149)(96 138)(97 127)(98 156)(99 145)(100 134)(101 123)(102 152)(103 141)(104 130)(105 159)(106 148)(107 137)(108 126)(109 155)(110 144)(111 133)(112 122)(113 151)(114 140)(115 129)(116 158)(117 147)(118 136)(119 125)(120 154)

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

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

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

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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