Copied to
clipboard

G = M4(2).1C8order 128 = 27

1st non-split extension by M4(2) of C8 acting via C8/C4=C2

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

Aliases: M4(2).1C8, C23.13M4(2), M5(2).23C22, C8.5(C2×C8), C8.39(C4⋊C4), C4.20(C4⋊C8), (C2×C8).33Q8, C8.27(C2×Q8), C8⋊C4.7C4, (C2×C8).393D4, C8.137(C2×D4), C8.C810C2, C4.29(C22×C8), C22.14(C4⋊C8), C42.170(C2×C4), (C4×C8).157C22, (C2×C8).606C23, (C2×C4).26M4(2), C42⋊C2.22C4, C82M4(2).6C2, (C2×M5(2)).26C2, (C2×M4(2)).14C4, (C22×C8).418C22, C22.24(C2×M4(2)), C2.16(C2×C4⋊C8), C4.82(C2×C4⋊C4), (C2×C8).88(C2×C4), (C2×C4).27(C2×C8), (C2×C4).140(C4⋊C4), (C2×C4).561(C22×C4), (C22×C4).288(C2×C4), SmallGroup(128,885)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2).1C8
C1C2C4C8C2×C8C22×C8C82M4(2) — M4(2).1C8
C1C2C4 — M4(2).1C8
C1C8C22×C8 — M4(2).1C8
C1C2C2C2C2C4C4C2×C8 — M4(2).1C8

Generators and relations for M4(2).1C8
 G = < a,b,c | a8=b2=1, c8=a4, bab=a5, cac-1=a-1, cbc-1=a4b >

Subgroups: 100 in 76 conjugacy classes, 58 normal (18 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22, C8 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], C23, C16 [×4], C42 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×6], M4(2) [×4], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C2×C16 [×2], M5(2) [×4], M5(2) [×2], C42⋊C2, C22×C8, C2×M4(2), C8.C8 [×4], C82M4(2), C2×M5(2) [×2], M4(2).1C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, M4(2).1C8

Smallest permutation representation of M4(2).1C8
On 32 points
Generators in S32
(1 30 13 26 9 22 5 18)(2 19 6 23 10 27 14 31)(3 32 15 28 11 24 7 20)(4 21 8 25 12 29 16 17)
(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)
(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)| (1,30,13,26,9,22,5,18)(2,19,6,23,10,27,14,31)(3,32,15,28,11,24,7,20)(4,21,8,25,12,29,16,17), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31), (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( (1,30,13,26,9,22,5,18)(2,19,6,23,10,27,14,31)(3,32,15,28,11,24,7,20)(4,21,8,25,12,29,16,17), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31), (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([(1,30,13,26,9,22,5,18),(2,19,6,23,10,27,14,31),(3,32,15,28,11,24,7,20),(4,21,8,25,12,29,16,17)], [(1,9),(3,11),(5,13),(7,15),(17,25),(19,27),(21,29),(23,31)], [(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)])

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E···8J8K8L8M8N16A···16P
order1222244444444488888···8888816···16
size1122211222444411112···244444···4

44 irreducible representations

dim1111111122224
type+++++-
imageC1C2C2C2C4C4C4C8D4Q8M4(2)M4(2)M4(2).1C8
kernelM4(2).1C8C8.C8C82M4(2)C2×M5(2)C8⋊C4C42⋊C2C2×M4(2)M4(2)C2×C8C2×C8C2×C4C23C1
# reps14124221622224

Matrix representation of M4(2).1C8 in GL4(𝔽17) generated by

0400
1000
0001
00130
,
16000
0100
0010
00016
,
0010
0001
8000
0800
G:=sub<GL(4,GF(17))| [0,1,0,0,4,0,0,0,0,0,0,13,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[0,0,8,0,0,0,0,8,1,0,0,0,0,1,0,0] >;

M4(2).1C8 in GAP, Magma, Sage, TeX

M_4(2)._1C_8
% in TeX

G:=Group("M4(2).1C8");
// GroupNames label

G:=SmallGroup(128,885);
// by ID

G=gap.SmallGroup(128,885);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,723,2019,248,102,124]);
// Polycyclic

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

׿
×
𝔽