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G = M4(2).13D6order 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, Q8.14D64C2, C12.196(C2×D4), D4.5(C3⋊D4), C36(D4.3D4), D4.Dic36C2, C12.53D46C2, C12.D410C2, C12.47D45C2, (C2×C12).15C23, Q8.12(C3⋊D4), C6.125(C4⋊D4), (C6×D4).105C22, C4.Dic3.25C22, C2.31(C23.14D6), C22.14(D42S3), (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

C1C2×C12 — M4(2).13D6
C1C3C6C12C2×C12C2×Dic6Q8.14D6 — M4(2).13D6
C3C6C2×C12 — M4(2).13D6
C1C2C2×C4C8⋊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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], C6, C6 [×3], C8 [×5], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8 [×2], C23, Dic3, C12 [×2], C12, C2×C6, C2×C6 [×3], C2×C8 [×2], M4(2), M4(2) [×3], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8 [×2], C3⋊C8 [×2], C24, Dic6 [×2], C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4 [×3], 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 [×2], C4.Dic3, D4.S3 [×3], 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 [×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, D4.3D4, 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 D42S3, 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)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)
(1 32 40 4 29 35)(2 31 33 3 30 34)(5 28 36 8 25 39)(6 27 37 7 26 38)(9 19 45 16 20 44)(10 18 46 15 21 43)(11 17 47 14 22 42)(12 24 48 13 23 41)
(1 47 3 41 5 43 7 45)(2 46 4 48 6 42 8 44)(9 40 11 34 13 36 15 38)(10 35 12 37 14 39 16 33)(17 28 19 30 21 32 23 26)(18 27 20 29 22 31 24 25)

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

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

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

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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