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G = C40.29D4order 320 = 26·5

29th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.29D4, D20.27D4, Dic10.27D4, C4.69(D4×D5), (C2×Q16)⋊10D5, (C10×Q16)⋊2C2, (C2×C8).97D10, C40.6C44C2, C20.190(C2×D4), C54(D4.5D4), C8.29(C5⋊D4), (C2×Q8).68D10, C20.10D48C2, (C2×C40).34C22, C2.24(C202D4), (C2×C20).466C23, D20.3C4.1C2, C20.C23.2C2, C4○D20.48C22, (Q8×C10).95C22, C10.123(C4⋊D4), C4.Dic5.21C22, C22.22(D42D5), C4.87(C2×C5⋊D4), (C2×C4).128(C22×D5), (C2×C10).160(C4○D4), SmallGroup(320,819)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.29D4
C1C5C10C20C2×C20C4○D20D20.3C4 — C40.29D4
C5C10C2×C20 — C40.29D4
C1C2C2×C4C2×Q16

Generators and relations for C40.29D4
 G = < a,b,c | a40=c2=1, b4=a20, bab-1=a-1, cac=a9, cbc=b3 >

Subgroups: 334 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], D5, C10, C10, C2×C8, C2×C8, M4(2) [×4], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], C52C8 [×3], C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20 [×2], C5×Q8 [×4], D4.5D4, C8×D5, C8⋊D5, C4.Dic5, C4.Dic5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×Q16 [×2], C4○D20, Q8×C10 [×2], C40.6C4, C20.10D4 [×2], D20.3C4, C20.C23 [×2], C10×Q16, C40.29D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4.5D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, C40.29D4

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C8D8E8F8G10A···10F20A20B20C20D20E···20L40A···40H
order12224444455888888810···102020202020···2040···40
size1122022882022224202040402···244448···84···4

44 irreducible representations

dim111111222222224444
type++++++++++++-+-
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10C5⋊D4D4.5D4D4×D5D42D5C40.29D4
kernelC40.29D4C40.6C4C20.10D4D20.3C4C20.C23C10×Q16C40Dic10D20C2×Q16C2×C10C2×C8C2×Q8C8C5C4C22C1
# reps112121211222482228

Matrix representation of C40.29D4 in GL4(𝔽41) generated by

22203121
003140
312427
19212119
,
317632
9201112
32241539
31402116
,
640640
3535035
00348
00357
G:=sub<GL(4,GF(41))| [22,0,31,19,20,0,2,21,31,31,4,21,21,40,27,19],[31,9,32,31,7,20,24,40,6,11,15,21,32,12,39,16],[6,35,0,0,40,35,0,0,6,0,34,35,40,35,8,7] >;

C40.29D4 in GAP, Magma, Sage, TeX

C_{40}._{29}D_4
% in TeX

G:=Group("C40.29D4");
// GroupNames label

G:=SmallGroup(320,819);
// by ID

G=gap.SmallGroup(320,819);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,184,1123,297,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^3>;
// generators/relations

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