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G = Q8.11D6order 96 = 25·3

1st non-split extension by Q8 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.19D4, Q8.11D6, C12.15C23, D12.10C22, Dic6.9C22, (C6×Q8)⋊2C2, (C2×Q8)⋊4S3, C3⋊Q165C2, C6.54(C2×D4), (C2×C4).20D6, (C2×C6).42D4, C3⋊C8.3C22, Q82S35C2, C4○D12.5C2, C34(C8.C22), C4.Dic37C2, C4.17(C3⋊D4), C4.15(C22×S3), (C3×Q8).6C22, (C2×C12).37C22, C22.11(C3⋊D4), C2.18(C2×C3⋊D4), SmallGroup(96,149)

Series: Derived Chief Lower central Upper central

C1C12 — Q8.11D6
C1C3C6C12D12C4○D12 — Q8.11D6
C3C6C12 — Q8.11D6
C1C2C2×C4C2×Q8

Generators and relations for Q8.11D6
 G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c5 >

Subgroups: 130 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], Dic3, C12 [×2], C12 [×2], D6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C8.C22, C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C4○D12, C6×Q8, Q8.11D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C3⋊D4 [×2], C22×S3, C8.C22, C2×C3⋊D4, Q8.11D6

Character table of Q8.11D6

 class 12A2B2C34A4B4C4D4E6A6B6C8A8B12A12B12C12D12E12F
 size 1121222244122221212444444
ρ1111111111111111111111    trivial
ρ2111-111111-1111-1-1111111    linear of order 2
ρ3111-1111-1-1-111111-1-1-1-111    linear of order 2
ρ411-1-11-111-11-11-11-1-1-1111-1    linear of order 2
ρ511-1-11-11-111-11-1-1111-1-11-1    linear of order 2
ρ61111111-1-11111-1-1-1-1-1-111    linear of order 2
ρ711-111-111-1-1-11-1-11-1-1111-1    linear of order 2
ρ811-111-11-11-1-11-11-111-1-11-1    linear of order 2
ρ92220-122220-1-1-100-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-20-1-222-201-110011-1-1-11    orthogonal lifted from D6
ρ1122202-2-2000222000000-2-2    orthogonal lifted from D4
ρ1222-2022-2000-22-2000000-22    orthogonal lifted from D4
ρ1322-20-1-22-2201-1100-1-111-11    orthogonal lifted from D6
ρ142220-122-2-20-1-1-1001111-1-1    orthogonal lifted from D6
ρ152220-1-2-2000-1-1-100-3--3--3-311    complex lifted from C3⋊D4
ρ1622-20-12-20001-1100-3--3-3--31-1    complex lifted from C3⋊D4
ρ1722-20-12-20001-1100--3-3--3-31-1    complex lifted from C3⋊D4
ρ182220-1-2-2000-1-1-100--3-3-3--311    complex lifted from C3⋊D4
ρ194-4004000000-4000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-400-200000-2-322-300000000    complex faithful
ρ214-400-2000002-32-2-300000000    complex faithful

Smallest permutation representation of Q8.11D6
On 48 points
Generators in S48
(1 23 7 17)(2 24 8 18)(3 13 9 19)(4 14 10 20)(5 15 11 21)(6 16 12 22)(25 44 31 38)(26 45 32 39)(27 46 33 40)(28 47 34 41)(29 48 35 42)(30 37 36 43)
(1 33 7 27)(2 28 8 34)(3 35 9 29)(4 30 10 36)(5 25 11 31)(6 32 12 26)(13 48 19 42)(14 43 20 37)(15 38 21 44)(16 45 22 39)(17 40 23 46)(18 47 24 41)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)(25 41 31 47)(26 46 32 40)(27 39 33 45)(28 44 34 38)(29 37 35 43)(30 42 36 48)

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

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

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

Q8.11D6 is a maximal subgroup of
D12.6D4  D12.7D4  C427D6  D12.15D4  C24.44D4  C24.29D4  D12.39D4  D12.40D4  SD1613D6  D12.30D4  S3×C8.C22  D24⋊C22  C12.C24  D12.34C23  D12.35C23  C36.C23  Q8.D18  D12.32D6  Dic6.29D6  D12.24D6  Dic6.22D6  C62.134D4  SL2(𝔽3).D6  D12.37D10  C12.D20  D12.27D10  C60.39C23  Q8.11D30
Q8.11D6 is a maximal quotient of
C4⋊C4.225D6  C4○D12⋊C4  (C2×C6).40D8  C4⋊C4.231D6  Q8.5Dic6  C42.56D6  Q8.6D12  C42.59D6  (C2×Q8).51D6  D12.37D4  C3⋊C86D4  C3⋊C8.6D4  C42.76D6  C42.77D6  C125SD16  C42.80D6  D126Q8  C42.82D6  C12⋊Q16  Dic66Q8  (C6×Q8)⋊6C4  (C3×Q8)⋊13D4  (C2×C6)⋊8Q16  C36.C23  D12.32D6  Dic6.29D6  D12.24D6  Dic6.22D6  C62.134D4  D12.37D10  C12.D20  D12.27D10  C60.39C23  Q8.11D30

Matrix representation of Q8.11D6 in GL4(𝔽7) generated by

6611
2041
3301
4351
,
1666
6036
1145
2562
,
0116
2514
1461
2223
,
0116
5502
6416
5211
G:=sub<GL(4,GF(7))| [6,2,3,4,6,0,3,3,1,4,0,5,1,1,1,1],[1,6,1,2,6,0,1,5,6,3,4,6,6,6,5,2],[0,2,1,2,1,5,4,2,1,1,6,2,6,4,1,3],[0,5,6,5,1,5,4,2,1,0,1,1,6,2,6,1] >;

Q8.11D6 in GAP, Magma, Sage, TeX

Q_8._{11}D_6
% in TeX

G:=Group("Q8.11D6");
// GroupNames label

G:=SmallGroup(96,149);
// by ID

G=gap.SmallGroup(96,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,86,579,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=1,b^2=c^6=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;
// generators/relations

Export

Character table of Q8.11D6 in TeX

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