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G = Q165C8order 128 = 27

5th semidirect product of Q16 and C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q165C8, C42.636C23, C8.4(C2×C8), C2.8(C8×D4), Q8⋊C8.1C2, (C8×Q8).3C2, Q8.2(C2×C8), C8⋊C8.1C2, C82C8.5C2, (C2×C8).305D4, C2.D8.18C4, C4.11(C8○D4), C4.11(C22×C8), (C4×C8).12C22, (C2×Q16).11C4, (C4×Q16).15C2, C22.81(C4×D4), C2.2(C8.26D4), C4⋊C8.274C22, Q8⋊C4.10C4, C2.1(Q16⋊C4), (C4×Q8).258C22, C4.136(C8.C22), C4⋊C4.130(C2×C4), (C2×C8).124(C2×C4), (C2×C4).1472(C2×D4), (C2×Q8).134(C2×C4), (C2×C4).497(C4○D4), (C2×C4).328(C22×C4), SmallGroup(128,311)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — Q165C8
C1C2C22C2×C4C42C4×C8C8×Q8 — Q165C8
C1C2C4 — Q165C8
C1C2×C4C4×C8 — Q165C8
C1C22C22C42 — Q165C8

Generators and relations for Q165C8
 G = < a,b,c | a8=c8=1, b2=a4, bab-1=a-1, cac-1=a5, bc=cb >

Subgroups: 120 in 80 conjugacy classes, 50 normal (22 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×7], C22, C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×4], Q8 [×4], Q8 [×2], C42, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×3], Q16 [×4], C2×Q8 [×2], C4×C8 [×3], C4×C8 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C8⋊C8, Q8⋊C8 [×2], C82C8, C4×Q16, C8×Q8 [×2], Q165C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8.C22 [×2], C8×D4, Q16⋊C4, C8.26D4, Q165C8

Smallest permutation representation of Q165C8
Regular action on 128 points
Generators in S128
(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)
(1 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 106 21 110)(18 105 22 109)(19 112 23 108)(20 111 24 107)(41 58 45 62)(42 57 46 61)(43 64 47 60)(44 63 48 59)(49 76 53 80)(50 75 54 79)(51 74 55 78)(52 73 56 77)(65 82 69 86)(66 81 70 85)(67 88 71 84)(68 87 72 83)(89 116 93 120)(90 115 94 119)(91 114 95 118)(92 113 96 117)(97 124 101 128)(98 123 102 127)(99 122 103 126)(100 121 104 125)
(1 105 49 72 128 120 14 59)(2 110 50 69 121 117 15 64)(3 107 51 66 122 114 16 61)(4 112 52 71 123 119 9 58)(5 109 53 68 124 116 10 63)(6 106 54 65 125 113 11 60)(7 111 55 70 126 118 12 57)(8 108 56 67 127 115 13 62)(17 75 86 104 92 37 47 29)(18 80 87 101 93 34 48 26)(19 77 88 98 94 39 41 31)(20 74 81 103 95 36 42 28)(21 79 82 100 96 33 43 25)(22 76 83 97 89 38 44 30)(23 73 84 102 90 35 45 27)(24 78 85 99 91 40 46 32)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I···4P8A···8H8I···8X
order1222444444444···48···88···8
size1111111122224···42···24···4

44 irreducible representations

dim111111111122244
type+++++++-
imageC1C2C2C2C2C2C4C4C4C8D4C4○D4C8○D4C8.C22C8.26D4
kernelQ165C8C8⋊C8Q8⋊C8C82C8C4×Q16C8×Q8Q8⋊C4C2.D8C2×Q16Q16C2×C8C2×C4C4C4C2
# reps1121124221622422

Matrix representation of Q165C8 in GL6(𝔽17)

740000
13100000
0013113
001613141
00412416
005414
,
0160000
1600000
00216612
0016151211
00103216
00371615
,
900000
090000
000010
000001
0013000
0001300

G:=sub<GL(6,GF(17))| [7,13,0,0,0,0,4,10,0,0,0,0,0,0,13,16,4,5,0,0,1,13,12,4,0,0,1,14,4,1,0,0,3,1,16,4],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,2,16,10,3,0,0,16,15,3,7,0,0,6,12,2,16,0,0,12,11,16,15],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0] >;

Q165C8 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_5C_8
% in TeX

G:=Group("Q16:5C8");
// GroupNames label

G:=SmallGroup(128,311);
// by ID

G=gap.SmallGroup(128,311);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=c^8=1,b^2=a^4,b*a*b^-1=a^-1,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations

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