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G = M4(2).D6order 192 = 26·3

12nd non-split extension by M4(2) of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).12D6, C3⋊C8.30D4, C8⋊C222S3, D4⋊D64C2, C4○D4.40D6, (C3×D4).11D4, (C2×D4).79D6, C4.178(S3×D4), (C3×Q8).11D4, C12.D49C2, C12.195(C2×D4), D4.4(C3⋊D4), C35(D4.4D4), D4.Dic35C2, C12.46D46C2, C12.53D45C2, (C2×C12).14C23, Q8.11(C3⋊D4), C6.124(C4⋊D4), (C6×D4).104C22, (C2×D12).129C22, (C3×M4(2)).9C22, C4.Dic3.24C22, C2.30(C23.14D6), C22.13(D42S3), (C2×D4⋊S3)⋊22C2, (C3×C8⋊C22)⋊6C2, C4.51(C2×C3⋊D4), (C2×C6).36(C4○D4), (C2×C3⋊C8).170C22, (C2×C4).14(C22×S3), (C3×C4○D4).12C22, SmallGroup(192,758)

Series: Derived Chief Lower central Upper central

C1C2×C12 — M4(2).D6
C1C3C6C12C2×C12C2×D12D4⋊D6 — M4(2).D6
C3C6C2×C12 — M4(2).D6
C1C2C2×C4C8⋊C22

Generators and relations for M4(2).D6
 G = < a,b,c,d | a8=b2=c6=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 336 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3, C6, C6 [×3], C8 [×5], C2×C4, C2×C4, D4, D4 [×5], Q8, C23 [×2], C12 [×2], C12, D6 [×2], C2×C6, C2×C6 [×3], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×4], SD16 [×2], C2×D4, C2×D4, C4○D4, C3⋊C8 [×2], C3⋊C8 [×2], C24, D12 [×2], C2×C12, C2×C12, C3×D4, C3×D4 [×3], C3×Q8, C22×S3, C22×C6, C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, D4⋊S3 [×3], Q82S3, C3×M4(2), C3×D8, C3×SD16, C2×D12, C6×D4, C3×C4○D4, D4.4D4, C12.53D4, C12.46D4, C12.D4, C2×D4⋊S3, D4.Dic3, D4⋊D6, C3×C8⋊C22, M4(2).D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.4D4, C23.14D6, M4(2).D6

Character table of M4(2).D6

 class 12A2B2C2D2E34A4B4C6A6B6C6D6E8A8B8C8D8E8F8G12A12B12C24A24B
 size 11248242224248886681212122444888
ρ1111111111111111111111111111    trivial
ρ2111-1-1-1111-111-1-1-1111-11-1111-111    linear of order 2
ρ3111-11-1111-11111-1-1-1-11-11111-1-1-1    linear of order 2
ρ41111-11111111-1-11-1-1-1-1-1-11111-1-1    linear of order 2
ρ5111-111111-11111-111-1-11-1-111-1-1-1    linear of order 2
ρ61111-1-1111111-1-1111-1111-1111-1-1    linear of order 2
ρ711111-1111111111-1-11-1-1-1-111111    linear of order 2
ρ8111-1-11111-111-1-1-1-1-111-11-111-111    linear of order 2
ρ9222220-1222-1-1-1-1-10020000-1-1-1-1-1    orthogonal lifted from S3
ρ102222-20-1222-1-111-100-20000-1-1-111    orthogonal lifted from D6
ρ11222-220-122-2-1-1-1-1100-20000-1-1111    orthogonal lifted from D6
ρ1222-200022-202-2000-2-2002002-2000    orthogonal lifted from D4
ρ1322-200022-202-20002200-2002-2000    orthogonal lifted from D4
ρ1422-2-2002-2222-200-20000000-22200    orthogonal lifted from D4
ρ15222-2-20-122-2-1-11110020000-1-11-1-1    orthogonal lifted from D6
ρ1622-22002-22-22-20020000000-22-200    orthogonal lifted from D4
ρ1722-2200-1-22-2-11-3--3-100000001-11-3--3    complex lifted from C3⋊D4
ρ1822-2-200-1-222-11--3-3100000001-1-1-3--3    complex lifted from C3⋊D4
ρ1922-2-200-1-222-11-3--3100000001-1-1--3-3    complex lifted from C3⋊D4
ρ2022-2200-1-22-2-11--3-3-100000001-11--3-3    complex lifted from C3⋊D4
ρ212220002-2-2022000000-2i02i0-2-2000    complex lifted from C4○D4
ρ222220002-2-20220000002i0-2i0-2-2000    complex lifted from C4○D4
ρ2344-4000-24-40-220000000000-22000    orthogonal lifted from S3×D4
ρ244-400004000-40000-22220000000000    orthogonal lifted from D4.4D4
ρ254-400004000-4000022-220000000000    orthogonal lifted from D4.4D4
ρ26444000-2-4-40-2-2000000000022000    symplectic lifted from D42S3, Schur index 2
ρ278-80000-400040000000000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of M4(2).D6
On 48 points
Generators in S48
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)
(1 16 46 22 38 29)(2 15 47 21 39 28)(3 14 48 20 40 27)(4 13 41 19 33 26)(5 12 42 18 34 25)(6 11 43 17 35 32)(7 10 44 24 36 31)(8 9 45 23 37 30)
(1 29 7 27 5 25 3 31)(2 28 8 26 6 32 4 30)(9 39 15 37 13 35 11 33)(10 38 16 36 14 34 12 40)(17 41 23 47 21 45 19 43)(18 48 24 46 22 44 20 42)

G:=sub<Sym(48)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (1,16,46,22,38,29)(2,15,47,21,39,28)(3,14,48,20,40,27)(4,13,41,19,33,26)(5,12,42,18,34,25)(6,11,43,17,35,32)(7,10,44,24,36,31)(8,9,45,23,37,30), (1,29,7,27,5,25,3,31)(2,28,8,26,6,32,4,30)(9,39,15,37,13,35,11,33)(10,38,16,36,14,34,12,40)(17,41,23,47,21,45,19,43)(18,48,24,46,22,44,20,42)>;

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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (1,16,46,22,38,29)(2,15,47,21,39,28)(3,14,48,20,40,27)(4,13,41,19,33,26)(5,12,42,18,34,25)(6,11,43,17,35,32)(7,10,44,24,36,31)(8,9,45,23,37,30), (1,29,7,27,5,25,3,31)(2,28,8,26,6,32,4,30)(9,39,15,37,13,35,11,33)(10,38,16,36,14,34,12,40)(17,41,23,47,21,45,19,43)(18,48,24,46,22,44,20,42) );

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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48)], [(1,16,46,22,38,29),(2,15,47,21,39,28),(3,14,48,20,40,27),(4,13,41,19,33,26),(5,12,42,18,34,25),(6,11,43,17,35,32),(7,10,44,24,36,31),(8,9,45,23,37,30)], [(1,29,7,27,5,25,3,31),(2,28,8,26,6,32,4,30),(9,39,15,37,13,35,11,33),(10,38,16,36,14,34,12,40),(17,41,23,47,21,45,19,43),(18,48,24,46,22,44,20,42)])

Matrix representation of M4(2).D6 in GL6(𝔽73)

0720000
7200000
004021032
0040214132
00161600
0015332112
,
7200000
0720000
0072000
0007200
000010
00391501
,
36280000
28360000
000010
003458171
001000
001394615
,
37280000
45360000
003458171
000010
001000
003114639

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,72,0,0,0,0,0,0,0,40,40,16,15,0,0,21,21,16,33,0,0,0,41,0,21,0,0,32,32,0,12],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,39,0,0,0,72,0,15,0,0,0,0,1,0,0,0,0,0,0,1],[36,28,0,0,0,0,28,36,0,0,0,0,0,0,0,34,1,1,0,0,0,58,0,39,0,0,1,1,0,46,0,0,0,71,0,15],[37,45,0,0,0,0,28,36,0,0,0,0,0,0,34,0,1,31,0,0,58,0,0,1,0,0,1,1,0,46,0,0,71,0,0,39] >;

M4(2).D6 in GAP, Magma, Sage, TeX

M_4(2).D_6
% in TeX

G:=Group("M4(2).D6");
// GroupNames label

G:=SmallGroup(192,758);
// by ID

G=gap.SmallGroup(192,758);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,1123,297,136,1684,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

Export

Character table of M4(2).D6 in TeX

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