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G = Q8.15D6order 96 = 25·3

1st non-split extension by Q8 of D6 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8.15D6, C6.10C24, D6.5C23, C312- 1+4, C12.24C23, D12.13C22, Dic3.6C23, Dic6.13C22, (C2×Q8)⋊7S3, (C6×Q8)⋊7C2, (S3×Q8)⋊4C2, Q8(C3⋊D4), C4○D126C2, (C2×C4).24D6, Q83S34C2, (C4×S3).5C22, C2.11(S3×C23), (C2×C6).68C23, C4.24(C22×S3), C3⋊D4.2C22, (C2×C12).48C22, C22.7(C22×S3), (C3×Q8).10C22, SmallGroup(96,214)

Series: Derived Chief Lower central Upper central

C1C6 — Q8.15D6
C1C3C6D6C4×S3S3×Q8 — Q8.15D6
C3C6 — Q8.15D6
C1C2C2×Q8

Generators and relations for Q8.15D6
 G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c5 >

Subgroups: 274 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2 [×5], C3, C4 [×6], C4 [×4], C22, C22 [×4], S3 [×4], C6, C6, C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], Dic3 [×4], C12 [×6], D6 [×4], C2×C6, C2×Q8, C2×Q8 [×4], C4○D4 [×10], Dic6 [×6], C4×S3 [×12], D12 [×6], C3⋊D4 [×4], C2×C12 [×3], C3×Q8 [×4], 2- 1+4, C4○D12 [×6], S3×Q8 [×4], Q83S3 [×4], C6×Q8, Q8.15D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2- 1+4, S3×C23, Q8.15D6

Character table of Q8.15D6

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G4H4I4J6A6B6C12A12B12C12D12E12F
 size 112666622222226666222444444
ρ1111111111111111111111111111    trivial
ρ211-111-111-11-1-111-1-1-11-1-11111-1-1-1    linear of order 2
ρ3111-1-1-1-11111111-1-1-1-1111111111    linear of order 2
ρ411-1-1-11-11-11-1-111111-1-1-11111-1-1-1    linear of order 2
ρ5111-1-1111-111-1-1-1-11-111111-1-1-1-11    linear of order 2
ρ611-1-1-1-11111-11-1-11-111-1-111-1-111-1    linear of order 2
ρ711111-1-11-111-1-1-11-11-11111-1-1-1-11    linear of order 2
ρ811-1111-1111-11-1-1-11-1-1-1-111-1-111-1    linear of order 2
ρ911-1-11-1-111-11-11-111-11-1-11-1-111-11    linear of order 2
ρ10111-111-11-1-1-111-1-1-111111-1-11-11-1    linear of order 2
ρ1111-11-11111-11-11-1-1-11-1-1-11-1-111-11    linear of order 2
ρ121111-1-111-1-1-111-111-1-1111-1-11-11-1    linear of order 2
ρ1311-11-1-1-11-1-111-11-1111-1-11-11-1-111    linear of order 2
ρ141111-11-111-1-1-1-111-1-11111-11-11-1-1    linear of order 2
ρ1511-1-11111-1-111-111-1-1-1-1-11-11-1-111    linear of order 2
ρ16111-11-1111-1-1-1-11-111-1111-11-11-1-1    linear of order 2
ρ1722-20000-122-22-2-2000011-1-111-1-11    orthogonal lifted from D6
ρ182220000-12222220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ192220000-12-2-2-2-220000-1-1-11-11-111    orthogonal lifted from D6
ρ2022-20000-12-22-22-2000011-111-1-11-1    orthogonal lifted from D6
ρ2122-20000-1-2-222-22000011-11-111-1-1    orthogonal lifted from D6
ρ222220000-1-2-2-222-20000-1-1-111-11-11    orthogonal lifted from D6
ρ232220000-1-222-2-2-20000-1-1-1-11111-1    orthogonal lifted from D6
ρ2422-20000-1-22-2-222000011-1-1-1-1111    orthogonal lifted from D6
ρ254-4000004000000000000-4000000    symplectic lifted from 2- 1+4, Schur index 2
ρ264-400000-20000000000-2-32-32000000    complex faithful
ρ274-400000-200000000002-3-2-32000000    complex faithful

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

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

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

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

Q8.15D6 is a maximal subgroup of
D12.4D4  D12.5D4  D12.14D4  D12.15D4  SD1613D6  D12.30D4  C24.C23  SD16.D6  C6.C25  S3×2- 1+4  D12.39C23  Q8.15D18  SL2(𝔽3).11D6  D12.33D6  D12.25D6  Dic6.26D6  C3272- 1+4  C30.C24  C30.33C24  D12.29D10  Q8.15D30
Q8.15D6 is a maximal quotient of
C6.72+ 1+4  C6.82+ 1+4  C6.2- 1+4  C6.2+ 1+4  C6.52- 1+4  C6.62- 1+4  C42.122D6  Q86Dic6  C42.125D6  C42.126D6  Q86D12  C42.132D6  C42.133D6  C42.134D6  C6.152- 1+4  C6.162- 1+4  C6.172- 1+4  D1222D4  Dic622D4  C6.202- 1+4  C6.212- 1+4  C6.222- 1+4  C6.232- 1+4  C6.242- 1+4  C6.252- 1+4  C6.592+ 1+4  C42.147D6  C42.150D6  C42.151D6  C42.154D6  C42.157D6  C42.158D6  Dic68Q8  C42.171D6  D1212D4  D128Q8  C42.174D6  C42.176D6  C42.177D6  C42.178D6  C42.180D6  C6.422- 1+4  Q8×C3⋊D4  C6.442- 1+4  C6.452- 1+4  Q8.15D18  D12.33D6  D12.25D6  Dic6.26D6  C3272- 1+4  C30.C24  C30.33C24  D12.29D10  Q8.15D30

Matrix representation of Q8.15D6 in GL4(𝔽7) generated by

4366
0545
1043
2641
,
6364
5316
2263
5646
,
3624
2036
2122
1252
,
3002
3325
0244
2004
G:=sub<GL(4,GF(7))| [4,0,1,2,3,5,0,6,6,4,4,4,6,5,3,1],[6,5,2,5,3,3,2,6,6,1,6,4,4,6,3,6],[3,2,2,1,6,0,1,2,2,3,2,5,4,6,2,2],[3,3,0,2,0,3,2,0,0,2,4,0,2,5,4,4] >;

Q8.15D6 in GAP, Magma, Sage, TeX

Q_8._{15}D_6
% in TeX

G:=Group("Q8.15D6");
// GroupNames label

G:=SmallGroup(96,214);
// by ID

G=gap.SmallGroup(96,214);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,86,579,2309]);
// 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=d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations

Export

Character table of Q8.15D6 in TeX

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