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G = Q8oD12order 96 = 25·3

Central product of Q8 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8oD12, D4oDic6, D4.10D6, Q8.16D6, C6.13C24, D6.8C23, C3:22- 1+4, C12.27C23, D12.14C22, Dic3.8C23, Dic6.14C22, C4oD4:6S3, (S3xQ8):5C2, C4oD12:9C2, (C2xC4).25D6, C3:D4.C22, D4:2S3:5C2, (C2xC6).5C23, (C2xDic6):14C2, (C4xS3).6C22, C4.34(C22xS3), C2.14(S3xC23), (C2xC12).49C22, (C3xD4).10C22, C22.4(C22xS3), (C3xQ8).11C22, (C2xDic3).22C22, (C3xC4oD4):5C2, SmallGroup(96,217)

Series: Derived Chief Lower central Upper central

C1C6 — Q8oD12
C1C3C6D6C4xS3S3xQ8 — Q8oD12
C3C6 — Q8oD12
C1C2C4oD4

Generators and relations for Q8oD12
 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, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2xC6, C2xQ8, C4oD4, C4oD4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C3xQ8, 2- 1+4, C2xDic6, C4oD12, D4:2S3, S3xQ8, C3xC4oD4, Q8oD12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2- 1+4, S3xC23, Q8oD12

Character table of Q8oD12

 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 Q8oD12
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 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)
(1 46 7 40)(2 47 8 41)(3 48 9 42)(4 37 10 43)(5 38 11 44)(6 39 12 45)(13 25 19 31)(14 26 20 32)(15 27 21 33)(16 28 22 34)(17 29 23 35)(18 30 24 36)
(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 45)(38 44)(39 43)(40 42)(46 48)

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

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

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

Q8oD12 is a maximal subgroup of
Q8.14D12  D4.10D12  D12.38D4  D12.40D4  D4.11D12  D4.13D12  D8:11D6  D8.10D6  D8:6D6  SD16.D6  D12.33C23  D12.35C23  C6.C25  D6.C24  S3x2- 1+4  D4.10D18  D12.A4  D12.33D6  D12.34D6  Dic6.24D6  D12.25D6  C32:92- 1+4  D20.38D6  D20.39D6  C15:2- 1+4  D20.29D6  D4.10D30
Q8oD12 is a maximal quotient of
C42.87D6  C42.89D6  C42.90D6  C42.91D6  C42.92D6  C42.94D6  C42.96D6  C42.98D6  C42.99D6  D4xDic6  C42.105D6  C42.106D6  C42.108D6  D12:24D4  Dic6:23D4  D4:6D12  C42.115D6  C42.118D6  Dic6:10Q8  Q8:7Dic6  C42.125D6  Q8xD12  C42.134D6  C42.135D6  C6.322+ 1+4  Dic6:19D4  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  Dic6:21D4  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  Dic6:10D4  C42.144D6  C42.145D6  Dic6:7Q8  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  C32:92- 1+4  D20.38D6  D20.39D6  C15:2- 1+4  D20.29D6  D4.10D30

Matrix representation of Q8oD12 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] >;

Q8oD12 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 Q8oD12 in TeX

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