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G = M4(2)⋊9D4  order 128 = 27

3rd semidirect product of M4(2) and D4 acting via D4/C4=C2

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

Aliases: M4(2)⋊9D4, C42.250D4, C42.375C23, C8.2(C2×D4), C8⋊3D4⋊4C2, (C4×C8)⋊26C22, C8.2D4⋊4C2, C4⋊Q8⋊68C22, (C4×M4(2))⋊6C2, C4.11(C22×D4), C8⋊C4⋊47C22, C8.12D4⋊14C2, C4.40(C4⋊1D4), C4⋊1D4⋊39C22, (C2×C8).269C23, (C2×C4).351C24, (C2×Q16)⋊20C22, (C2×D8).62C22, C23.685(C2×D4), (C22×C4).470D4, (C2×SD16)⋊16C22, (C2×D4).117C23, C4.4D4⋊59C22, (C2×Q8).105C23, C22.17(C4⋊1D4), (C2×C42).857C22, C22.611(C22×D4), C2.39(D8⋊C22), (C22×C4).1041C23, C22.26C24⋊11C2, (C22×D4).377C22, (C22×Q8).310C22, (C2×M4(2)).271C22, (C2×C8⋊C22)⋊23C2, (C2×C4).695(C2×D4), C2.30(C2×C4⋊1D4), (C2×C4.4D4)⋊42C2, (C2×C8.C22)⋊23C2, (C2×C4○D4).157C22, SmallGroup(128,1885)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — M4(2)⋊9D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — M4(2)⋊9D4
Lower central
C1 — C2 — C2×C4 — M4(2)⋊9D4
Upper central
C1 — C22 — C2×C42 — M4(2)⋊9D4
Jennings
C1 — C2 — C2 — C2×C4 — M4(2)⋊9D4

Generators and relations for M4(2)⋊9D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 596 in 286 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, C2×C42, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C8⋊C22, C8.C22, C22×D4, C22×Q8, C2×C4○D4, C4×M4(2), C8.12D4, C8⋊3D4, C8.2D4, C2×C4.4D4, C22.26C24, C2×C8⋊C22, C2×C8.C22, M4(2)⋊9D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C22×D4, C2×C4⋊1D4, D8⋊C22, M4(2)⋊9D4

Smallest permutation representation of M4(2)⋊9D4
►On 32 points
Generators in S32
(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)

(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I4J4K4L4M4N8A···8H
order12222222224···44444448···8
size11112288882···24488884···4

32 irreducible representations

dim1111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D8⋊C22
kernelM4(2)⋊9D4C4×M4(2)C8.12D4C8⋊3D4C8.2D4C2×C4.4D4C22.26C24C2×C8⋊C22C2×C8.C22C42M4(2)C22×C4C2
# reps1142211222824

Matrix representation of M4(2)⋊9D4 ►in GL6(𝔽17)

010000
1600000
000040
001313139
0001300
000404
,
1600000
0160000
001000
000100
0000160
001616016
,
010000
1600000
004000
000400
0000130
001313013
,
1600000
010000
0016161615
000010
000100
000001

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,13,13,4,0,0,4,13,0,0,0,0,0,9,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,13,0,0,0,4,0,13,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,1,0,0,0,16,1,0,0,0,0,15,0,0,1] >;
 

M4(2)⋊9D4 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_9D_4
 
% in TeX
 
G:=Group("M4(2):9D4");
 
// GroupNames label
 
G:=SmallGroup(128,1885);
 
// by ID
 
G=gap.SmallGroup(128,1885);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,520,1018,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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