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G = M4(2)⋊Q8  order 128 = 27

1st semidirect product of M4(2) and Q8 acting via Q8/C2=C22

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

Aliases: C42⋊2Q8, M4(2)⋊1Q8, C4⋊C4.102D4, C4.24(C4⋊Q8), (C22×C4).85D4, C4.9(C22⋊Q8), C42⋊6C4.2C2, C23.593(C2×D4), C22.13(C4⋊Q8), C2.35(D4⋊4D4), C42⋊9C4.14C2, C22.230C22≀C2, C22.C42.3C2, M4(2)⋊C4.5C2, (C2×C42).374C22, (C22×C4).733C23, C22.35(C22⋊Q8), C2.29(D4.10D4), C42⋊C2.64C22, (C2×M4(2)).25C22, C23.41C23.5C2, C2.7(C23.78C23), (C2×C4).18(C2×Q8), (C2×C4).1051(C2×D4), (C2×C4).348(C4○D4), (C2×C4⋊C4).137C22, SmallGroup(128,792)

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

Derived series
C1 — C22×C4 — M4(2)⋊Q8
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C42⋊9C4 — M4(2)⋊Q8
Lower central
C1 — C2 — C22×C4 — M4(2)⋊Q8
Upper central
C1 — C22 — C22×C4 — M4(2)⋊Q8
Jennings
C1 — C2 — C2 — C22×C4 — M4(2)⋊Q8

Generators and relations for M4(2)⋊Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a5, cac-1=a-1, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=c-1 >

Subgroups: 248 in 123 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C42⋊6C4, C22.C42, C42⋊9C4, M4(2)⋊C4, C23.41C23, M4(2)⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4⋊Q8, C23.78C23, D4⋊4D4, D4.10D4, M4(2)⋊Q8

Character table of M4(2)⋊Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-11-1-1-1-111111    linear of order 2
ρ31111111111-1-1-1-111-1-1-111-1-1-111    linear of order 2
ρ411111111111111-1-1-111-1-1-1-1-111    linear of order 2
ρ51111111111-1-1-1-11-11111-11-1-1-1-1    linear of order 2
ρ611111111111111-111-1-1-111-1-1-1-1    linear of order 2
ρ7111111111111111-1-1-1-11-1-111-1-1    linear of order 2
ρ81111111111-1-1-1-1-11-111-11-111-1-1    linear of order 2
ρ92222-2-22-2-220000000-220000000    orthogonal lifted from D4
ρ10222222-2-2-2-2000000-2000020000    orthogonal lifted from D4
ρ112222-2-2-222-20000-200002000000    orthogonal lifted from D4
ρ12222222-2-2-2-200000020000-20000    orthogonal lifted from D4
ρ132222-2-22-2-2200000002-20000000    orthogonal lifted from D4
ρ142222-2-2-222-2000020000-2000000    orthogonal lifted from D4
ρ152-2-22-22-22-22000000000000-2200    symplectic lifted from Q8, Schur index 2
ρ162-2-22-22-22-220000000000002-200    symplectic lifted from Q8, Schur index 2
ρ172-2-222-222-2-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ182-2-22-222-22-2000000000000002-2    symplectic lifted from Q8, Schur index 2
ρ192-2-22-222-22-200000000000000-22    symplectic lifted from Q8, Schur index 2
ρ202-2-222-222-2-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ212-2-222-2-2-22200000-2i00002i00000    complex lifted from C4○D4
ρ222-2-222-2-2-222000002i0000-2i00000    complex lifted from C4○D4
ρ234-44-400000022-2-2000000000000    orthogonal lifted from D4⋊4D4
ρ244-44-4000000-2-222000000000000    orthogonal lifted from D4⋊4D4
ρ2544-4-40000002-2-22000000000000    symplectic lifted from D4.10D4, Schur index 2
ρ2644-4-4000000-222-2000000000000    symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of M4(2)⋊Q8
►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)

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

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

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

Matrix representation of M4(2)⋊Q8 ►in GL6(𝔽17)

6150000
10110000
000010
000001
000100
0016000
,
100000
010000
001000
000100
0000160
0000016
,
120000
16160000
00001311
0000114
00131100
0011400
,
1120000
760000
00131100
0011400
00001311
0000114

G:=sub<GL(6,GF(17))| [6,10,0,0,0,0,15,11,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,0,13,11,0,0,0,0,11,4,0,0,13,11,0,0,0,0,11,4,0,0],[11,7,0,0,0,0,2,6,0,0,0,0,0,0,13,11,0,0,0,0,11,4,0,0,0,0,0,0,13,11,0,0,0,0,11,4] >;
 

M4(2)⋊Q8 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes Q_8
 
% in TeX
 
G:=Group("M4(2):Q8");
 
// GroupNames label
 
G:=SmallGroup(128,792);
 
// by ID
 
G=gap.SmallGroup(128,792);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,58,2804,1411,718,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of M4(2)⋊Q8 in TeX

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