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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, C4○D4.42D6, (C3×D4).15D4, C4.180(S3×D4), C8.C22⋊2S3, (C3×Q8).15D4, (C2×Q8).93D6, D4⋊D6.2C2, C12.199(C2×D4), D4.6(C3⋊D4), C3⋊7(D4.3D4), D4.Dic3⋊7C2, C12.46D4⋊8C2, C12.10D4⋊9C2, C12.53D4⋊7C2, (C2×C12).18C23, Q8.13(C3⋊D4), (C6×Q8).96C22, C6.126(C4⋊D4), (C2×D12).131C22, C4.Dic3.27C22, C2.32(C23.14D6), C22.15(D4⋊2S3), (C3×M4(2)).12C22, C4.55(C2×C3⋊D4), (C3×C8.C22)⋊6C2, (C2×Q8⋊2S3)⋊22C2, (C2×C6).38(C4○D4), (C2×C3⋊C8).172C22, (C2×C4).18(C22×S3), (C3×C4○D4).16C22, SmallGroup(192,762)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — M4(2).15D6
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — D4⋊D6 — M4(2).15D6
Lower central
C3 — C6 — C2×C12 — M4(2).15D6
Upper central
C1 — C2 — C2×C4 — 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, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, 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, Q8⋊2S3, C3×M4(2), C3×SD16, C3×Q16, C2×D12, C6×Q8, C3×C4○D4, D4.3D4, C12.53D4, C12.46D4, C12.10D4, C2×Q8⋊2S3, D4.Dic3, D4⋊D6, C3×C8.C22, M4(2).15D6
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).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 C4○D4
ρ22222002-2-200220000-2i2i00-2-200000    complex lifted from C4○D4
ρ2344-400-24-400-2200000000-2200000    orthogonal lifted from S3×D4
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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