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G = Q8○D12order 96 = 25·3

Central product of Q8 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8D12, D4Dic6, D4.10D6, Q8.16D6, C6.13C24, D6.8C23, C322- 1+4, C12.27C23, D12.14C22, Dic3.8C23, Dic6.14C22, C4○D46S3, (S3×Q8)⋊5C2, C4○D129C2, (C2×C4).25D6, C3⋊D4.C22, D42S35C2, (C2×C6).5C23, (C2×Dic6)⋊14C2, (C4×S3).6C22, C4.34(C22×S3), C2.14(S3×C23), (C2×C12).49C22, (C3×D4).10C22, C22.4(C22×S3), (C3×Q8).11C22, (C2×Dic3).22C22, (C3×C4○D4)⋊5C2, SmallGroup(96,217)

Series: Derived Chief Lower central Upper central

C1C6 — Q8○D12
C1C3C6D6C4×S3S3×Q8 — Q8○D12
C3C6 — Q8○D12
C1C2C4○D4

Generators and relations for Q8○D12
 G = < a,b,c,d | a4=d2=1, b2=c6=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c5 >

Subgroups: 266 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3 [×2], C6, C6 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×9], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, 2- 1+4, C2×Dic6 [×3], C4○D12 [×3], D42S3 [×6], S3×Q8 [×2], C3×C4○D4, Q8○D12
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2- 1+4, S3×C23, Q8○D12

Character table of Q8○D12

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G4H4I4J6A6B6C6D12A12B12C12D12E
 size 112226622222666666244422444
ρ1111111111111111111111111111    trivial
ρ211111-1-111111-1-1-1-1-1-1111111111    linear of order 2
ρ3111-1-11111-11-1-1-1-111-111-1-111-1-11    linear of order 2
ρ4111-1-1-1-111-11-1111-1-1111-1-111-1-11    linear of order 2
ρ511-1-11111-111-11-1-1-1-111-1-1111-11-1    linear of order 2
ρ611-1-11-1-11-111-1-11111-11-1-1111-11-1    linear of order 2
ρ711-11-1111-1-111-111-1-1-11-11-1111-1-1    linear of order 2
ρ811-11-1-1-11-1-1111-1-11111-11-1111-1-1    linear of order 2
ρ91111-1-111-11-1-1-1-11-111111-1-1-1-11-1    linear of order 2
ρ101111-11-11-11-1-111-11-1-1111-1-1-1-11-1    linear of order 2
ρ11111-11-111-1-1-1111-1-11-111-11-1-11-1-1    linear of order 2
ρ12111-111-11-1-1-11-1-111-1111-11-1-11-1-1    linear of order 2
ρ1311-1-1-1-11111-11-11-11-111-1-1-1-1-1111    linear of order 2
ρ1411-1-1-11-1111-111-11-11-11-1-1-1-1-1111    linear of order 2
ρ1511-111-1111-1-1-11-111-1-11-111-1-1-1-11    linear of order 2
ρ1611-1111-111-1-1-1-11-1-1111-111-1-1-1-11    linear of order 2
ρ1722-22-200-1-2-222000000-11-11-1-1-111    orthogonal lifted from D6
ρ182222200-12222000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1922-2-2200-1-222-2000000-111-1-1-11-11    orthogonal lifted from D6
ρ20222-2-200-12-22-2000000-1-111-1-111-1    orthogonal lifted from D6
ρ2122-2-2-200-122-22000000-111111-1-1-1    orthogonal lifted from D6
ρ22222-2200-1-2-2-22000000-1-11-111-111    orthogonal lifted from D6
ρ2322-22200-12-2-2-2000000-11-1-11111-1    orthogonal lifted from D6
ρ242222-200-1-22-2-2000000-1-1-11111-11    orthogonal lifted from D6
ρ254-40000040000000000-400000000    symplectic lifted from 2- 1+4, Schur index 2
ρ264-400000-200000000002000-2323000    symplectic faithful, Schur index 2
ρ274-400000-20000000000200023-23000    symplectic faithful, Schur index 2

Smallest permutation representation of Q8○D12
On 48 points
Generators in S48
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)
(1 37 7 43)(2 38 8 44)(3 39 9 45)(4 40 10 46)(5 41 11 47)(6 42 12 48)(13 31 19 25)(14 32 20 26)(15 33 21 27)(16 34 22 28)(17 35 23 29)(18 36 24 30)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 27)(28 36)(29 35)(30 34)(31 33)(37 39)(40 48)(41 47)(42 46)(43 45)

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

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

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

Q8○D12 is a maximal subgroup of
Q8.14D12  D4.10D12  D12.38D4  D12.40D4  D4.11D12  D4.13D12  D811D6  D8.10D6  D86D6  SD16.D6  D12.33C23  D12.35C23  C6.C25  D6.C24  S3×2- 1+4  D4.10D18  D12.A4  D12.33D6  D12.34D6  Dic6.24D6  D12.25D6  C3292- 1+4  D20.38D6  D20.39D6  C15⋊2- 1+4  D20.29D6  D4.10D30
Q8○D12 is a maximal quotient of
C42.87D6  C42.89D6  C42.90D6  C42.91D6  C42.92D6  C42.94D6  C42.96D6  C42.98D6  C42.99D6  D4×Dic6  C42.105D6  C42.106D6  C42.108D6  D1224D4  Dic623D4  D46D12  C42.115D6  C42.118D6  Dic610Q8  Q87Dic6  C42.125D6  Q8×D12  C42.134D6  C42.135D6  C6.322+ 1+4  Dic619D4  C6.702- 1+4  C6.712- 1+4  C6.722- 1+4  C6.732- 1+4  C6.492+ 1+4  C6.752- 1+4  C6.152- 1+4  C6.162- 1+4  Dic621D4  C6.772- 1+4  C6.782- 1+4  C6.252- 1+4  C6.792- 1+4  C6.802- 1+4  C6.812- 1+4  C6.822- 1+4  C6.632+ 1+4  C6.652+ 1+4  C6.852- 1+4  C6.692+ 1+4  C42.137D6  C42.139D6  C42.140D6  C42.141D6  Dic610D4  C42.144D6  C42.145D6  Dic67Q8  C42.147D6  C42.148D6  C42.152D6  C42.154D6  C42.156D6  C42.157D6  C42.159D6  C42.160D6  C42.161D6  C42.162D6  C42.164D6  C42.165D6  C6.1042- 1+4  C6.1052- 1+4  C6.1442+ 1+4  C6.1072- 1+4  C6.1082- 1+4  D4.10D18  D12.33D6  D12.34D6  Dic6.24D6  D12.25D6  C3292- 1+4  D20.38D6  D20.39D6  C15⋊2- 1+4  D20.29D6  D4.10D30

Matrix representation of Q8○D12 in GL4(𝔽13) generated by

8000
0800
5850
10505
,
8011
08111
0050
0005
,
10300
10700
0073
001010
,
1100
01200
00121
0001
G:=sub<GL(4,GF(13))| [8,0,5,10,0,8,8,5,0,0,5,0,0,0,0,5],[8,0,0,0,0,8,0,0,1,11,5,0,1,1,0,5],[10,10,0,0,3,7,0,0,0,0,7,10,0,0,3,10],[1,0,0,0,1,12,0,0,0,0,12,0,0,0,1,1] >;

Q8○D12 in GAP, Magma, Sage, TeX

Q_8\circ D_{12}
% in TeX

G:=Group("Q8oD12");
// GroupNames label

G:=SmallGroup(96,217);
// by ID

G=gap.SmallGroup(96,217);
# by ID

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

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

Export

Character table of Q8○D12 in TeX

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