Copied to
clipboard

G = C42.9Q8order 128 = 27

9th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.9Q8, C42.369D4, C4⋊C83C4, C4.38C4≀C2, C8⋊C411C4, C22.12C4≀C2, C42.38(C2×C4), (C2×C4).12C42, C424C4.4C2, (C22×C4).642D4, (C4×M4(2)).12C2, C42.6C4.9C2, C2.12(C426C4), C2.C42.12C4, (C2×C42).135C22, C2.8(M4(2)⋊4C4), C23.142(C22⋊C4), C22.50(C2.C42), (C2×C4).21(C4⋊C4), (C22×C4).155(C2×C4), (C2×C4).303(C22⋊C4), SmallGroup(128,32)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.9Q8
C1C2C22C23C22×C4C2×C42C424C4 — C42.9Q8
C1C22C2×C4 — C42.9Q8
C1C2×C4C2×C42 — C42.9Q8
C1C22C22C2×C42 — C42.9Q8

Generators and relations for C42.9Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=ab2c-1 >

Subgroups: 160 in 86 conjugacy classes, 34 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×13], C23, C42 [×4], C42 [×4], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2.C42 [×2], C2.C42, C4×C8, C8⋊C4 [×2], C8⋊C4, C22⋊C8, C4⋊C8 [×2], C2×C42, C2×C42, C2×M4(2), C424C4, C4×M4(2), C42.6C4, C42.9Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4≀C2 [×4], C426C4 [×2], M4(2)⋊4C4, C42.9Q8

Smallest permutation representation of C42.9Q8
On 32 points
Generators in S32
(1 17 31 14)(2 22 32 11)(3 19 25 16)(4 24 26 13)(5 21 27 10)(6 18 28 15)(7 23 29 12)(8 20 30 9)
(1 7 5 3)(2 26 6 30)(4 28 8 32)(9 22 13 18)(10 16 14 12)(11 24 15 20)(17 23 21 19)(25 31 29 27)
(1 7 5 3)(2 20 32 9)(4 22 26 11)(6 24 28 13)(8 18 30 15)(10 12 14 16)(17 19 21 23)(25 31 29 27)
(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)

G:=sub<Sym(32)| (1,17,31,14)(2,22,32,11)(3,19,25,16)(4,24,26,13)(5,21,27,10)(6,18,28,15)(7,23,29,12)(8,20,30,9), (1,7,5,3)(2,26,6,30)(4,28,8,32)(9,22,13,18)(10,16,14,12)(11,24,15,20)(17,23,21,19)(25,31,29,27), (1,7,5,3)(2,20,32,9)(4,22,26,11)(6,24,28,13)(8,18,30,15)(10,12,14,16)(17,19,21,23)(25,31,29,27), (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)>;

G:=Group( (1,17,31,14)(2,22,32,11)(3,19,25,16)(4,24,26,13)(5,21,27,10)(6,18,28,15)(7,23,29,12)(8,20,30,9), (1,7,5,3)(2,26,6,30)(4,28,8,32)(9,22,13,18)(10,16,14,12)(11,24,15,20)(17,23,21,19)(25,31,29,27), (1,7,5,3)(2,20,32,9)(4,22,26,11)(6,24,28,13)(8,18,30,15)(10,12,14,16)(17,19,21,23)(25,31,29,27), (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) );

G=PermutationGroup([(1,17,31,14),(2,22,32,11),(3,19,25,16),(4,24,26,13),(5,21,27,10),(6,18,28,15),(7,23,29,12),(8,20,30,9)], [(1,7,5,3),(2,26,6,30),(4,28,8,32),(9,22,13,18),(10,16,14,12),(11,24,15,20),(17,23,21,19),(25,31,29,27)], [(1,7,5,3),(2,20,32,9),(4,22,26,11),(6,24,28,13),(8,18,30,15),(10,12,14,16),(17,19,21,23),(25,31,29,27)], [(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)])

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K···4T8A···8H8I8J8K8L
order12222244444···44···48···88888
size11112211112···24···44···48888

38 irreducible representations

dim1111111222224
type+++++-+
imageC1C2C2C2C4C4C4D4Q8D4C4≀C2C4≀C2M4(2)⋊4C4
kernelC42.9Q8C424C4C4×M4(2)C42.6C4C2.C42C8⋊C4C4⋊C8C42C42C22×C4C4C22C2
# reps1111444112882

Matrix representation of C42.9Q8 in GL4(𝔽17) generated by

01600
1000
00130
00013
,
13000
01300
00160
0001
,
13000
0400
00160
0004
,
61100
111100
0001
00160
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,13,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,4],[6,11,0,0,11,11,0,0,0,0,0,16,0,0,1,0] >;

C42.9Q8 in GAP, Magma, Sage, TeX

C_4^2._9Q_8
% in TeX

G:=Group("C4^2.9Q8");
// GroupNames label

G:=SmallGroup(128,32);
// by ID

G=gap.SmallGroup(128,32);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924]);
// Polycyclic

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

׿
×
𝔽