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G = M4(2).13D6  order 192 = 26·3

13rd non-split extension by M4(2) of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).13D6, C3⋊C8.31D4, C8⋊C22.S3, C4○D4.41D6, (C3×D4).12D4, (C2×D4).80D6, C4.179(S3×D4), (C3×Q8).12D4, Q8.14D6⋊4C2, C12.196(C2×D4), D4.5(C3⋊D4), C3⋊6(D4.3D4), D4.Dic3⋊6C2, C12.53D4⋊6C2, C12.D4⋊10C2, C12.47D4⋊5C2, (C2×C12).15C23, Q8.12(C3⋊D4), C6.125(C4⋊D4), (C6×D4).105C22, C4.Dic3.25C22, C2.31(C23.14D6), C22.14(D4⋊2S3), (C2×Dic6).134C22, (C3×M4(2)).10C22, C4.52(C2×C3⋊D4), (C2×D4.S3)⋊22C2, (C3×C8⋊C22).1C2, (C2×C6).37(C4○D4), (C2×C3⋊C8).171C22, (C2×C4).15(C22×S3), (C3×C4○D4).13C22, SmallGroup(192,759)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — M4(2).13D6
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — Q8.14D6 — M4(2).13D6
Lower central
C3 — C6 — C2×C12 — M4(2).13D6
Upper central
C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for M4(2).13D6
 G = < a,b,c,d | a8=b2=c6=1, d2=a2, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 272 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4.S3, C3⋊Q16, C3×M4(2), C3×D8, C3×SD16, C2×Dic6, C6×D4, C3×C4○D4, D4.3D4, C12.53D4, C12.47D4, C12.D4, C2×D4.S3, D4.Dic3, Q8.14D6, C3×C8⋊C22, M4(2).13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.3D4, C23.14D6, M4(2).13D6

Character table of M4(2).13D6

 class 12A2B2C2D34A4B4C4D6A6B6C6D6E8A8B8C8D8E8F8G12A12B12C24A24B
 size 11248222424248886681212122444888
ρ1111111111111111111111111111    trivial
ρ2111-11111-1111-11111-1-1-11-111-1-1-1    linear of order 2
ρ31111-11111-1111-1-111-1111-1111-1-1    linear of order 2
ρ4111-1-1111-1-111-1-1-1111-1-11111-111    linear of order 2
ρ5111111111-111111-1-11-1-1-1-111111    linear of order 2
ρ6111-11111-1-111-111-1-1-111-1111-1-1-1    linear of order 2
ρ71111-111111111-1-1-1-1-1-1-1-11111-1-1    linear of order 2
ρ8111-1-1111-1111-1-1-1-1-1111-1-111-111    linear of order 2
ρ922222-12220-1-1-1-1-10020000-1-1-1-1-1    orthogonal lifted from S3
ρ1022-2002-22002-200022000-20-22000    orthogonal lifted from D4
ρ11222-2-2-122-20-1-11110020000-1-11-1-1    orthogonal lifted from D6
ρ1222-2002-22002-2000-2-200020-22000    orthogonal lifted from D4
ρ1322-2-2022-2202-2-20000000002-2200    orthogonal lifted from D4
ρ142222-2-12220-1-1-11100-20000-1-1-111    orthogonal lifted from D6
ρ1522-22022-2-202-220000000002-2-200    orthogonal lifted from D4
ρ16222-22-122-20-1-11-1-100-20000-1-1111    orthogonal lifted from D6
ρ1722-2-20-12-220-111-√-3√-30000000-11-1√-3-√-3    complex lifted from C3⋊D4
ρ1822-220-12-2-20-11-1√-3-√-30000000-111√-3-√-3    complex lifted from C3⋊D4
ρ1922-2-20-12-220-111√-3-√-30000000-11-1-√-3√-3    complex lifted from C3⋊D4
ρ2022-220-12-2-20-11-1-√-3√-30000000-111-√-3√-3    complex lifted from C3⋊D4
ρ21222002-2-20022000000-2i2i00-2-2000    complex lifted from C4○D4
ρ22222002-2-200220000002i-2i00-2-2000    complex lifted from C4○D4
ρ2344-400-2-4400-2200000000002-2000    orthogonal lifted from S3×D4
ρ2444400-2-4-400-2-2000000000022000    symplectic lifted from D4⋊2S3, Schur index 2
ρ254-400040000-40000-2√-22√-20000000000    complex lifted from D4.3D4
ρ264-400040000-400002√-2-2√-20000000000    complex lifted from D4.3D4
ρ278-8000-4000040000000000000000    symplectic faithful, Schur index 2

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

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

Matrix representation of M4(2).13D6 ►in GL8(𝔽73)

00100000
00010000
720000000
072000000
000000720
000072727248
000007200
00003331
,
10000000
01000000
00100000
00010000
000072000
000007200
00000010
00003301
,
000720000
001720000
072000000
172000000
00000010
000072727248
00001000
00000001
,
5571000000
5318000000
001820000
0020550000
000066124
00006604
0000676700
00000555561

G:=sub<GL(8,GF(73))| [0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,3,0,0,0,0,0,72,72,3,0,0,0,0,72,72,0,3,0,0,0,0,0,48,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,3,0,0,0,0,0,72,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,48,0,1],[55,53,0,0,0,0,0,0,71,18,0,0,0,0,0,0,0,0,18,20,0,0,0,0,0,0,2,55,0,0,0,0,0,0,0,0,6,6,67,0,0,0,0,0,6,6,67,55,0,0,0,0,12,0,0,55,0,0,0,0,4,4,0,61] >;
 

M4(2).13D6 in GAP, Magma, Sage, TeX 

M_4(2)._{13}D_6
 
% in TeX
 
G:=Group("M4(2).13D6");
 
// GroupNames label
 
G:=SmallGroup(192,759);
 
// by ID
 
G=gap.SmallGroup(192,759);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,1123,297,136,1684,851,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^2,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).13D6 in TeX

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