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G = M4(2).15D6order 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.C222S3, (C3×Q8).15D4, (C2×Q8).93D6, D4⋊D6.2C2, C12.199(C2×D4), D4.6(C3⋊D4), C37(D4.3D4), D4.Dic37C2, C12.46D48C2, C12.10D49C2, C12.53D47C2, (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(D42S3), (C3×M4(2)).12C22, C4.55(C2×C3⋊D4), (C3×C8.C22)⋊6C2, (C2×Q82S3)⋊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

C1C2×C12 — M4(2).15D6
C1C3C6C12C2×C12C2×D12D4⋊D6 — M4(2).15D6
C3C6C2×C12 — M4(2).15D6
C1C2C2×C4C8.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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3, C6, C6 [×2], C8 [×5], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8 [×2], C23, C12 [×2], C12 [×2], D6 [×2], C2×C6, C2×C6, 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, D12 [×2], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C22×S3, 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, Q82S3 [×3], C3×M4(2), C3×SD16, C3×Q16, C2×D12, C6×Q8, C3×C4○D4, D4.3D4, C12.53D4, C12.46D4, C12.10D4, C2×Q82S3, D4.Dic3, D4⋊D6, C3×C8.C22, M4(2).15D6
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).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 D42S3, 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)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)
(1 38 46 14 31 21)(2 33 47 9 32 24)(3 36 48 12 25 19)(4 39 41 15 26 22)(5 34 42 10 27 17)(6 37 43 13 28 20)(7 40 44 16 29 23)(8 35 45 11 30 18)
(1 21 3 23 5 17 7 19)(2 24 4 18 6 20 8 22)(9 41 11 43 13 45 15 47)(10 44 12 46 14 48 16 42)(25 40 27 34 29 36 31 38)(26 35 28 37 30 39 32 33)

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

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

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

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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