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G = C426C8order 128 = 27

3rd semidirect product of C42 and C8 acting via C8/C4=C2

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

Aliases: C426C8, C43.5C2, C42.41Q8, C42.453D4, C4⋊C85C4, C4.1(C4×C8), C4.36C4≀C2, C4.22(C4⋊C8), C4.1(C8⋊C4), (C2×C42).41C4, (C2×C4).49C42, C42.250(C2×C4), (C22×C4).634D4, (C2×C4).65M4(2), C2.1(C426C4), C22.9(C22⋊C8), C42.12C4.3C2, C23.134(C22⋊C4), (C2×C42).1024C22, C2.5(C22.7C42), C22.17(C2.C42), (C2×C4).68(C2×C8), (C2×C4).155(C4⋊C4), (C22×C4).462(C2×C4), (C2×C4).368(C22⋊C4), SmallGroup(128,8)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C426C8
C1C2C22C23C22×C4C2×C42C43 — C426C8
C1C2C4 — C426C8
C1C42C2×C42 — C426C8
C1C22C22C2×C42 — C426C8

Generators and relations for C426C8
 G = < a,b,c | a4=b4=c8=1, cac-1=ab=ba, cbc-1=b-1 >

Subgroups: 168 in 104 conjugacy classes, 48 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×2], C42 [×6], C42 [×10], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C2×C42 [×2], C2×C42 [×2], C43, C42.12C4 [×2], C426C8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4≀C2 [×4], C4⋊C8 [×2], C22.7C42, C426C4 [×2], C426C8

Smallest permutation representation of C426C8
On 32 points
Generators in S32
(2 22 32 15)(4 24 26 9)(6 18 28 11)(8 20 30 13)
(1 21 31 14)(2 15 32 22)(3 23 25 16)(4 9 26 24)(5 17 27 10)(6 11 28 18)(7 19 29 12)(8 13 30 20)
(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)| (2,22,32,15)(4,24,26,9)(6,18,28,11)(8,20,30,13), (1,21,31,14)(2,15,32,22)(3,23,25,16)(4,9,26,24)(5,17,27,10)(6,11,28,18)(7,19,29,12)(8,13,30,20), (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( (2,22,32,15)(4,24,26,9)(6,18,28,11)(8,20,30,13), (1,21,31,14)(2,15,32,22)(3,23,25,16)(4,9,26,24)(5,17,27,10)(6,11,28,18)(7,19,29,12)(8,13,30,20), (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([(2,22,32,15),(4,24,26,9),(6,18,28,11),(8,20,30,13)], [(1,21,31,14),(2,15,32,22),(3,23,25,16),(4,9,26,24),(5,17,27,10),(6,11,28,18),(7,19,29,12),(8,13,30,20)], [(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)])

56 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4AH8A···8P
order1222224···44···48···8
size1111221···12···24···4

56 irreducible representations

dim11111122222
type++++-+
imageC1C2C2C4C4C8D4Q8D4M4(2)C4≀C2
kernelC426C8C43C42.12C4C4⋊C8C2×C42C42C42C42C22×C4C2×C4C4
# reps1128416112416

Matrix representation of C426C8 in GL4(𝔽17) generated by

1400
01600
0010
0004
,
16000
01600
0040
00013
,
9000
4800
0001
00160
G:=sub<GL(4,GF(17))| [1,0,0,0,4,16,0,0,0,0,1,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[9,4,0,0,0,8,0,0,0,0,0,16,0,0,1,0] >;

C426C8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_6C_8
% in TeX

G:=Group("C4^2:6C8");
// GroupNames label

G:=SmallGroup(128,8);
// by ID

G=gap.SmallGroup(128,8);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,184,248,1684]);
// Polycyclic

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

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