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G = M4(2).C8order 128 = 27

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

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

Aliases: M5(2)⋊6C4, M4(2).2C8, (C2×C16)⋊4C4, C8.35(C4⋊C4), C4.16(C4⋊C8), (C2×C8).28Q8, (C2×C8).375D4, C22⋊C4.2C8, C22.8(C4×C8), C23.6(C2×C8), C4.8(C8⋊C4), (C2×C4).55C42, C8.56(C22⋊C4), C4.21(C22⋊C8), C22.10(C4⋊C8), (C2×M5(2)).9C2, (C2×C4).12M4(2), C42⋊C2.13C4, (C2×M4(2)).19C4, C82M4(2).12C2, C22.24(C22⋊C8), (C22×C8).369C22, C4.28(C2.C42), C2.16(C22.7C42), (C2×C4).14(C2×C8), (C2×C8).238(C2×C4), (C2×C4).106(C4⋊C4), (C22×C4).166(C2×C4), (C2×C4).348(C22⋊C4), SmallGroup(128,110)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — M4(2).C8
C1C2C4C2×C4C2×C8C22×C8C82M4(2) — M4(2).C8
C1C2C22 — M4(2).C8
C1C8C22×C8 — M4(2).C8
C1C2C2C2C2C4C4C22×C8 — M4(2).C8

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

Subgroups: 96 in 66 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×2], C22 [×3], C22, C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×2], C23, C16 [×4], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×6], C2×C8, M4(2) [×2], M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16 [×2], C2×C16, M5(2) [×2], M5(2) [×3], C42⋊C2, C22×C8, C2×M4(2), C82M4(2), C2×M5(2) [×2], M4(2).C8
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⋊C8 [×2], C22.7C42, M4(2).C8

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

dim1111111112224
type++++-
imageC1C2C2C4C4C4C4C8C8D4Q8M4(2)M4(2).C8
kernelM4(2).C8C82M4(2)C2×M5(2)C2×C16M5(2)C42⋊C2C2×M4(2)C22⋊C4M4(2)C2×C8C2×C8C2×C4C1
# reps1124422883144

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

01500
2000
0020
00015
,
0100
1000
0001
0010
,
0010
0001
8000
0800
G:=sub<GL(4,GF(17))| [0,2,0,0,15,0,0,0,0,0,2,0,0,0,0,15],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,0,8,0,0,0,0,8,1,0,0,0,0,1,0,0] >;

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

M_4(2).C_8
% in TeX

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

G:=SmallGroup(128,110);
// by ID

G=gap.SmallGroup(128,110);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,723,136,2804,124]);
// 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*b,b*c=c*b>;
// generators/relations

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