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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — M4(2).D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — D4⋊D6 — M4(2).D6
 Lower central C3 — C6 — C2×C12 — M4(2).D6
 Upper central C1 — C2 — C2×C4 — C8⋊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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C3⋊C8, C3⋊C8, C24, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4⋊S3, 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, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.4D4, C23.14D6, M4(2).D6

Character table of M4(2).D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 24A 24B size 1 1 2 4 8 24 2 2 2 4 2 4 8 8 8 6 6 8 12 12 12 24 4 4 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 1 1 -1 1 1 linear of order 2 ρ3 1 1 1 -1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 1 linear of order 2 ρ8 1 1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 1 1 linear of order 2 ρ9 2 2 2 2 2 0 -1 2 2 2 -1 -1 -1 -1 -1 0 0 2 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -2 0 -1 2 2 2 -1 -1 1 1 -1 0 0 -2 0 0 0 0 -1 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 2 -2 2 0 -1 2 2 -2 -1 -1 -1 -1 1 0 0 -2 0 0 0 0 -1 -1 1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 0 0 0 2 2 -2 0 2 -2 0 0 0 -2 -2 0 0 2 0 0 2 -2 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 0 0 2 2 -2 0 2 -2 0 0 0 2 2 0 0 -2 0 0 2 -2 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 -2 0 0 2 -2 2 2 2 -2 0 0 -2 0 0 0 0 0 0 0 -2 2 2 0 0 orthogonal lifted from D4 ρ15 2 2 2 -2 -2 0 -1 2 2 -2 -1 -1 1 1 1 0 0 2 0 0 0 0 -1 -1 1 -1 -1 orthogonal lifted from D6 ρ16 2 2 -2 2 0 0 2 -2 2 -2 2 -2 0 0 2 0 0 0 0 0 0 0 -2 2 -2 0 0 orthogonal lifted from D4 ρ17 2 2 -2 2 0 0 -1 -2 2 -2 -1 1 √-3 -√-3 -1 0 0 0 0 0 0 0 1 -1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 0 0 -1 -2 2 2 -1 1 -√-3 √-3 1 0 0 0 0 0 0 0 1 -1 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ19 2 2 -2 -2 0 0 -1 -2 2 2 -1 1 √-3 -√-3 1 0 0 0 0 0 0 0 1 -1 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 -2 2 0 0 -1 -2 2 -2 -1 1 -√-3 √-3 -1 0 0 0 0 0 0 0 1 -1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ21 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 0 0 0 0 -2i 0 2i 0 -2 -2 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 0 0 2 -2 -2 0 2 2 0 0 0 0 0 0 2i 0 -2i 0 -2 -2 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 0 0 0 -2 4 -4 0 -2 2 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ25 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ26 4 4 4 0 0 0 -2 -4 -4 0 -2 -2 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ27 8 -8 0 0 0 0 -4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 35 46 22 11 29)(2 34 47 21 12 28)(3 33 48 20 13 27)(4 40 41 19 14 26)(5 39 42 18 15 25)(6 38 43 17 16 32)(7 37 44 24 9 31)(8 36 45 23 10 30)
(1 29 7 27 5 25 3 31)(2 28 8 26 6 32 4 30)(9 33 15 39 13 37 11 35)(10 40 16 38 14 36 12 34)(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), (1,35,46,22,11,29)(2,34,47,21,12,28)(3,33,48,20,13,27)(4,40,41,19,14,26)(5,39,42,18,15,25)(6,38,43,17,16,32)(7,37,44,24,9,31)(8,36,45,23,10,30), (1,29,7,27,5,25,3,31)(2,28,8,26,6,32,4,30)(9,33,15,39,13,37,11,35)(10,40,16,38,14,36,12,34)(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), (1,35,46,22,11,29)(2,34,47,21,12,28)(3,33,48,20,13,27)(4,40,41,19,14,26)(5,39,42,18,15,25)(6,38,43,17,16,32)(7,37,44,24,9,31)(8,36,45,23,10,30), (1,29,7,27,5,25,3,31)(2,28,8,26,6,32,4,30)(9,33,15,39,13,37,11,35)(10,40,16,38,14,36,12,34)(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)], [(1,35,46,22,11,29),(2,34,47,21,12,28),(3,33,48,20,13,27),(4,40,41,19,14,26),(5,39,42,18,15,25),(6,38,43,17,16,32),(7,37,44,24,9,31),(8,36,45,23,10,30)], [(1,29,7,27,5,25,3,31),(2,28,8,26,6,32,4,30),(9,33,15,39,13,37,11,35),(10,40,16,38,14,36,12,34),(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)

 0 72 0 0 0 0 72 0 0 0 0 0 0 0 40 21 0 32 0 0 40 21 41 32 0 0 16 16 0 0 0 0 15 33 21 12
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 39 15 0 1
,
 36 28 0 0 0 0 28 36 0 0 0 0 0 0 0 0 1 0 0 0 34 58 1 71 0 0 1 0 0 0 0 0 1 39 46 15
,
 37 28 0 0 0 0 45 36 0 0 0 0 0 0 34 58 1 71 0 0 0 0 1 0 0 0 1 0 0 0 0 0 31 1 46 39

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

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