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G = C42.47D10order 320 = 26·5

47th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.47D10, C52C828D4, C59(C89D4), (C4×D4).5D5, C203C821C2, (D4×C20).6C2, C4.215(D4×D5), C4⋊C4.7Dic5, C2.8(D4×Dic5), (D4×C10).28C4, C20.374(C2×D4), C10.121(C4×D4), (C2×C10)⋊7M4(2), (C2×D4).5Dic5, C10.63(C8○D4), (C4×C20).82C22, C42.D54C2, C22⋊C4.4Dic5, C20.307(C4○D4), C20.55D425C2, (C2×C20).849C23, (C22×C4).310D10, C10.74(C2×M4(2)), C2.6(D4.Dic5), C221(C4.Dic5), C4.134(D42D5), C23.17(C2×Dic5), (C22×C20).99C22, C22.45(C22×Dic5), (C5×C4⋊C4).24C4, (C2×C4.Dic5)⋊4C2, (C2×C20).336(C2×C4), C2.8(C2×C4.Dic5), (C22×C52C8)⋊19C2, (C5×C22⋊C4).13C4, (C2×C4).34(C2×Dic5), (C2×C4).791(C22×D5), (C22×C10).128(C2×C4), (C2×C10).287(C22×C4), (C2×C52C8).204C22, SmallGroup(320,638)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.47D10
C1C5C10C20C2×C20C2×C52C8C22×C52C8 — C42.47D10
C5C2×C10 — C42.47D10
C1C2×C4C4×D4

Generators and relations for C42.47D10
 G = < a,b,c,d | a4=b4=c10=1, d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 286 in 124 conjugacy classes, 61 normal (55 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×5], C2×C4 [×5], C2×C4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×5], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8 [×2], C52C8 [×3], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C89D4, C2×C52C8 [×4], C2×C52C8 [×2], C4.Dic5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, C42.D5, C203C8, C20.55D4 [×2], C22×C52C8, C2×C4.Dic5, D4×C20, C42.47D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×M4(2), C8○D4, C2×Dic5 [×6], C22×D5, C89D4, C4.Dic5 [×2], D4×D5, D42D5, C22×Dic5, C2×C4.Dic5, D4×Dic5, D4.Dic5, C42.47D10

Smallest permutation representation of C42.47D10
On 160 points
Generators in S160
(1 60 55 6)(2 7 56 51)(3 52 57 8)(4 9 58 53)(5 54 59 10)(11 16 21 26)(12 27 22 17)(13 18 23 28)(14 29 24 19)(15 20 25 30)(31 36 64 69)(32 70 65 37)(33 38 66 61)(34 62 67 39)(35 40 68 63)(41 126 158 109)(42 110 159 127)(43 128 160 101)(44 102 151 129)(45 130 152 103)(46 104 153 121)(47 122 154 105)(48 106 155 123)(49 124 156 107)(50 108 157 125)(71 84 89 76)(72 77 90 85)(73 86 81 78)(74 79 82 87)(75 88 83 80)(91 118 133 150)(92 141 134 119)(93 120 135 142)(94 143 136 111)(95 112 137 144)(96 145 138 113)(97 114 139 146)(98 147 140 115)(99 116 131 148)(100 149 132 117)
(1 73 63 22)(2 74 64 23)(3 75 65 24)(4 76 66 25)(5 77 67 26)(6 78 68 27)(7 79 69 28)(8 80 70 29)(9 71 61 30)(10 72 62 21)(11 54 90 39)(12 55 81 40)(13 56 82 31)(14 57 83 32)(15 58 84 33)(16 59 85 34)(17 60 86 35)(18 51 87 36)(19 52 88 37)(20 53 89 38)(41 116 121 136)(42 117 122 137)(43 118 123 138)(44 119 124 139)(45 120 125 140)(46 111 126 131)(47 112 127 132)(48 113 128 133)(49 114 129 134)(50 115 130 135)(91 155 145 101)(92 156 146 102)(93 157 147 103)(94 158 148 104)(95 159 149 105)(96 160 150 106)(97 151 141 107)(98 152 142 108)(99 153 143 109)(100 154 144 110)
(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)
(1 156 73 146 63 102 22 92)(2 155 74 145 64 101 23 91)(3 154 75 144 65 110 24 100)(4 153 76 143 66 109 25 99)(5 152 77 142 67 108 26 98)(6 151 78 141 68 107 27 97)(7 160 79 150 69 106 28 96)(8 159 80 149 70 105 29 95)(9 158 71 148 61 104 30 94)(10 157 72 147 62 103 21 93)(11 135 54 50 90 115 39 130)(12 134 55 49 81 114 40 129)(13 133 56 48 82 113 31 128)(14 132 57 47 83 112 32 127)(15 131 58 46 84 111 33 126)(16 140 59 45 85 120 34 125)(17 139 60 44 86 119 35 124)(18 138 51 43 87 118 36 123)(19 137 52 42 88 117 37 122)(20 136 53 41 89 116 38 121)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I5A5B8A···8H8I8J8K8L10A···10F10G···10N20A···20H20I···20X
order1222222444444444558···8888810···1010···1020···2020···20
size11112241111224442210···10202020202···24···42···24···4

68 irreducible representations

dim111111111122222222222444
type++++++++++--+-+-
imageC1C2C2C2C2C2C2C4C4C4D4D5C4○D4M4(2)D10Dic5Dic5D10Dic5C8○D4C4.Dic5D4×D5D42D5D4.Dic5
kernelC42.47D10C42.D5C203C8C20.55D4C22×C52C8C2×C4.Dic5D4×C20C5×C22⋊C4C5×C4⋊C4D4×C10C52C8C4×D4C20C2×C10C42C22⋊C4C4⋊C4C22×C4C2×D4C10C22C4C4C2
# reps1112111422222424242416224

Matrix representation of C42.47D10 in GL4(𝔽41) generated by

13900
14000
00400
0001
,
1000
0100
00320
00032
,
13900
04000
00230
00025
,
13900
04000
00025
00390
G:=sub<GL(4,GF(41))| [1,1,0,0,39,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,39,40,0,0,0,0,23,0,0,0,0,25],[1,0,0,0,39,40,0,0,0,0,0,39,0,0,25,0] >;

C42.47D10 in GAP, Magma, Sage, TeX

C_4^2._{47}D_{10}
% in TeX

G:=Group("C4^2.47D10");
// GroupNames label

G:=SmallGroup(320,638);
// by ID

G=gap.SmallGroup(320,638);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
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