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G = M4(2)⋊Q8order 128 = 27

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

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

Aliases: C422Q8, M4(2)⋊1Q8, C4⋊C4.102D4, C4.24(C4⋊Q8), (C22×C4).85D4, C4.9(C22⋊Q8), C426C4.2C2, C23.593(C2×D4), C22.13(C4⋊Q8), C2.35(D44D4), C429C4.14C2, 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

C1C22×C4 — M4(2)⋊Q8
C1C2C22C2×C4C22×C4C2×C4⋊C4C429C4 — M4(2)⋊Q8
C1C2C22×C4 — M4(2)⋊Q8
C1C22C22×C4 — M4(2)⋊Q8
C1C2C2C22×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 [×3], C2 [×2], C4 [×4], C4 [×11], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], Q8 [×2], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C426C4 [×2], C22.C42, C429C4, M4(2)⋊C4 [×2], C23.41C23, M4(2)⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, C2×D4 [×3], C2×Q8 [×3], C4○D4, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], C23.78C23, D44D4, 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 D44D4
ρ244-44-4000000-2-222000000000000    orthogonal lifted from D44D4
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 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)
(1 22 31 15)(2 21 32 14)(3 20 25 13)(4 19 26 12)(5 18 27 11)(6 17 28 10)(7 24 29 9)(8 23 30 16)
(1 23 31 16)(2 13 32 20)(3 21 25 14)(4 11 26 18)(5 19 27 12)(6 9 28 24)(7 17 29 10)(8 15 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)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23), (1,22,31,15)(2,21,32,14)(3,20,25,13)(4,19,26,12)(5,18,27,11)(6,17,28,10)(7,24,29,9)(8,23,30,16), (1,23,31,16)(2,13,32,20)(3,21,25,14)(4,11,26,18)(5,19,27,12)(6,9,28,24)(7,17,29,10)(8,15,30,22)>;

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

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

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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