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G = M4(2).15D6  order 192 = 26·3

15th non-split extension by M4(2) of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).15D6, C3:C8.32D4, C4oD4.42D6, (C3xD4).15D4, C4.180(S3xD4), C8.C22:2S3, (C3xQ8).15D4, (C2xQ8).93D6, D4:D6.2C2, C12.199(C2xD4), D4.6(C3:D4), C3:7(D4.3D4), D4.Dic3:7C2, C12.46D4:8C2, C12.10D4:9C2, C12.53D4:7C2, (C2xC12).18C23, Q8.13(C3:D4), (C6xQ8).96C22, C6.126(C4:D4), (C2xD12).131C22, C4.Dic3.27C22, C2.32(C23.14D6), C22.15(D4:2S3), (C3xM4(2)).12C22, C4.55(C2xC3:D4), (C3xC8.C22):6C2, (C2xQ8:2S3):22C2, (C2xC6).38(C4oD4), (C2xC3:C8).172C22, (C2xC4).18(C22xS3), (C3xC4oD4).16C22, SmallGroup(192,762)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — M4(2).15D6
Chief series
C1 — C3 — C6 — C12 — C2xC12 — C2xD12 — D4:D6 — M4(2).15D6
Lower central
C3 — C6 — C2xC12 — M4(2).15D6
Upper central
C1 — C2 — C2xC4 — C8.C22

Generators and relations for M4(2).15D6
 G = < a,b,c,d | a8=b2=c6=1, d2=a2, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >

Subgroups: 304 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C12, C12, D6, C2xC6, C2xC6, C2xC8, M4(2), M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, C3:C8, C3:C8, C24, D12, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xS3, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, C2xC3:C8, C2xC3:C8, C4.Dic3, C4.Dic3, D4:S3, Q8:2S3, C3xM4(2), C3xSD16, C3xQ16, C2xD12, C6xQ8, C3xC4oD4, D4.3D4, C12.53D4, C12.46D4, C12.10D4, C2xQ8:2S3, D4.Dic3, D4:D6, C3xC8.C22, M4(2).15D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, S3xD4, D4:2S3, C2xC3:D4, D4.3D4, C23.14D6, M4(2).15D6

Character table of M4(2).15D6

 class 12A2B2C2D34A4B4C4D6A6B6C8A8B8C8D8E8F8G12A12B12C12D12E24A24B
 size 11242422248248668121212244488888
ρ1111111111111111111111111111    trivial
ρ21111-11111-111111-1111-111-11-1-1-1    linear of order 2
ρ3111-11111-1111-111-1-1-11-1111-11-1-1    linear of order 2
ρ4111-1-1111-1-111-1111-1-11111-1-1-111    linear of order 2
ρ5111111111-1111-1-1-1-1-1-1111-11-1-1-1    linear of order 2
ρ61111-111111111-1-11-1-1-1-11111111    linear of order 2
ρ7111-11111-1-111-1-1-1111-1-111-1-1-111    linear of order 2
ρ8111-1-1111-1111-1-1-1-111-11111-11-1-1    linear of order 2
ρ922-20022-2002-20-2-2000202-200000    orthogonal lifted from D4
ρ10222-20-122-2-2-1-110020000-1-1111-1-1    orthogonal lifted from D6
ρ1122220-12222-1-1-10020000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-20022-2002-2022000-202-200000    orthogonal lifted from D4
ρ1322220-1222-2-1-1-100-20000-1-11-1111    orthogonal lifted from D6
ρ14222-20-122-22-1-1100-20000-1-1-11-111    orthogonal lifted from D6
ρ1522-2202-22-202-220000000-220-2000    orthogonal lifted from D4
ρ1622-2-202-22202-2-20000000-2202000    orthogonal lifted from D4
ρ1722-2-20-1-2220-11100000001-1√-3-1-√-3√-3-√-3    complex lifted from C3:D4
ρ1822-220-1-22-20-11-100000001-1-√-31√-3√-3-√-3    complex lifted from C3:D4
ρ1922-2-20-1-2220-11100000001-1-√-3-1√-3-√-3√-3    complex lifted from C3:D4
ρ2022-220-1-22-20-11-100000001-1√-31-√-3-√-3√-3    complex lifted from C3:D4
ρ21222002-2-2002200002i-2i00-2-200000    complex lifted from C4oD4
ρ22222002-2-200220000-2i2i00-2-200000    complex lifted from C4oD4
ρ2344-400-24-400-2200000000-2200000    orthogonal lifted from S3xD4
ρ2444400-2-4-400-2-2000000002200000    symplectic lifted from D4:2S3, Schur index 2
ρ254-400040000-4002√-2-2√-2000000000000    complex lifted from D4.3D4
ρ264-400040000-400-2√-22√-2000000000000    complex lifted from D4.3D4
ρ278-8000-4000040000000000000000    orthogonal faithful

Smallest permutation representation of M4(2).15D6
►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)

(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)

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

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

Matrix representation of M4(2).15D6 ►in GL8(Z)

00001-10-1
0000101-1
00000-1-11
00001-1-10
011-10000
-11100000
1-10-10000
101-10000
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
00000-100
00001-100
0000000-1
0000001-1
0-1000000
1-1000000
000-10000
001-10000
,
0000001-1
0000000-1
00001-100
00000-100
1-1000000
0-1000000
00-110000
00010000

G:=sub<GL(8,Integers())| [0,0,0,0,0,-1,1,1,0,0,0,0,1,1,-1,0,0,0,0,0,1,1,0,1,0,0,0,0,-1,0,-1,-1,1,1,0,1,0,0,0,0,-1,0,-1,-1,0,0,0,0,0,1,-1,-1,0,0,0,0,-1,-1,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-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,-1],[0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0] >;
 

M4(2).15D6 in GAP, Magma, Sage, TeX 

M_4(2)._{15}D_6
 
% in TeX
 
G:=Group("M4(2).15D6");
 
// GroupNames label
 
G:=SmallGroup(192,762);
 
// by ID
 
G=gap.SmallGroup(192,762);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,184,1123,297,136,1684,851,438,102,6278]);
 
// 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^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of M4(2).15D6 in TeX

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