Q128 - GroupNames

G = Q₁₂₈ order 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

Derived series

C₁ — C₃₂ — Q₁₂₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₁₆ — C₃₂ — Q₆₄ — Q₁₂₈

Lower central

C₁ — C₂ — C₄ — C₈ — C₁₆ — C₃₂ — Q₁₂₈

Upper central

C₁ — C₂ — C₄ — C₈ — C₁₆ — C₃₂ — Q₁₂₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₄ — C₄ — C₄ — C₄ — C₄ — C₄ — C₈ — C₈ — C₈ — C₈ — C₁₆ — C₁₆ — C₃₂ — Q₁₂₈

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

C₁

C₂

C₄

₁₆C₄

C₈

₈Q₈

C₁₆

₄Q₁₆

C₃₂

₂Q₃₂

C₆₄

Q₆₄

Q₁₂₈

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	4A	4B	4C	8A	8B	16A	16B	16C	16D	32A	···	32H	64A	···	64P
order	1	2	4	4	4	8	8	16	16	16	16	32	···	32	64	···	64
size	1	1	2	32	32	2	2	2	2	2	2	2	···	2	2	···	2

35 irreducible representations

dim	1	1	1	2	2	2	2	2
type	+	+	+	+	+	+	+	-
image	C₁	C₂	C₂	D₄	D₈	D₁₆	D₃₂	Q₁₂₈
kernel	Q₁₂₈	C₆₄	Q₆₄	C₁₆	C₈	C₄	C₂	C₁
# reps	1	1	2	1	2	4	8	16

Matrix representation of Q₁₂₈ ►in GL₂(𝔽₁₉₃) generated by

78	155
38	78

71	13
13	122

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

G = Q128 order 128 = 27

Generalised quaternion group

G = Q₁₂₈ order 128 = 2⁷