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G = Q128order 128 = 27

Generalised quaternion group

p-group, metacyclic, nilpotent (class 6), monomial

Aliases: Q128, Dic32, C64.C2, Q64.C2, C4.3D16, C16.7D4, C8.12D8, C2.5D32, C32.4C22, 2-Sylow(SL(2,191)), SmallGroup(128,163)

Series: Derived Chief Lower central Upper central Jennings

C1C32 — Q128
C1C2C4C8C16C32Q64 — Q128
C1C2C4C8C16C32 — Q128
C1C2C4C8C16C32 — Q128
C1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4C4C8C8C8C8C16C16C32 — Q128

Generators and relations for Q128
 G = < a,b | a64=1, b2=a32, bab-1=a-1 >

16C4
16C4
8Q8
8Q8
4Q16
4Q16
2Q32
2Q32

Smallest permutation representation of Q128
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 86 33 118)(2 85 34 117)(3 84 35 116)(4 83 36 115)(5 82 37 114)(6 81 38 113)(7 80 39 112)(8 79 40 111)(9 78 41 110)(10 77 42 109)(11 76 43 108)(12 75 44 107)(13 74 45 106)(14 73 46 105)(15 72 47 104)(16 71 48 103)(17 70 49 102)(18 69 50 101)(19 68 51 100)(20 67 52 99)(21 66 53 98)(22 65 54 97)(23 128 55 96)(24 127 56 95)(25 126 57 94)(26 125 58 93)(27 124 59 92)(28 123 60 91)(29 122 61 90)(30 121 62 89)(31 120 63 88)(32 119 64 87)

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

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

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

35 conjugacy classes

class 1  2 4A4B4C8A8B16A16B16C16D32A···32H64A···64P
order12444881616161632···3264···64
size11232322222222···22···2

35 irreducible representations

dim11122222
type+++++++-
imageC1C2C2D4D8D16D32Q128
kernelQ128C64Q64C16C8C4C2C1
# reps112124816

Matrix representation of Q128 in GL2(𝔽193) generated by

78155
3878
,
7113
13122
G:=sub<GL(2,GF(193))| [78,38,155,78],[71,13,13,122] >;

Q128 in GAP, Magma, Sage, TeX

Q_{128}
% in TeX

G:=Group("Q128");
// GroupNames label

G:=SmallGroup(128,163);
// by ID

G=gap.SmallGroup(128,163);
# by ID

G:=PCGroup([7,-2,2,-2,-2,-2,-2,-2,448,85,456,254,135,142,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b|a^64=1,b^2=a^32,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Q128 in TeX

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