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G = Q8.14D12  order 192 = 26·3

4th non-split extension by Q8 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.9D12, D12.34D4, Q8.14D12, C42.25D6, Dic6.34D4, M4(2).7D6, C4wrC2:3S3, (C3xD4).4D4, C12.5(C2xD4), (C3xQ8).4D4, C4oD4.19D6, C4.11(C2xD12), C8.D6:9C2, C4.127(S3xD4), Q8oD12.1C2, C42:4S3:9C2, C6.29C22wrC2, C12:2Q8:10C2, Q8.14D6:2C2, (C2xDic3).2D4, C22.31(S3xD4), C12.47D4:1C2, C3:2(D4.10D4), (C4xC12).52C22, C2.32(D6:D4), (C2xC12).266C23, C4oD12.15C22, (C2xDic6).76C22, (C3xM4(2)).4C22, C4.Dic3.10C22, (C3xC4wrC2):3C2, (C2xC6).28(C2xD4), (C3xC4oD4).7C22, (C2xC4).111(C22xS3), SmallGroup(192,385)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — Q8.14D12
Chief series
C1 — C3 — C6 — C12 — C2xC12 — C4oD12 — Q8oD12 — Q8.14D12
Lower central
C3 — C6 — C2xC12 — Q8.14D12
Upper central
C1 — C2 — C2xC4 — C4wrC2

Generators and relations for Q8.14D12
 G = < a,b,c,d | a4=1, b2=c12=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=c11 >

Subgroups: 416 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C4:C4, M4(2), M4(2), SD16, Q16, C2xQ8, C4oD4, C4oD4, C3:C8, C24, Dic6, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C4.10D4, C4wrC2, C4wrC2, C4:Q8, C8.C22, 2- 1+4, C24:C2, Dic12, C4.Dic3, C4:Dic3, D4.S3, C3:Q16, C4xC12, C3xM4(2), C2xDic6, C2xDic6, C4oD12, C4oD12, D4:2S3, S3xQ8, C3xC4oD4, D4.10D4, C42:4S3, C12.47D4, C3xC4wrC2, C12:2Q8, C8.D6, Q8.14D6, Q8oD12, Q8.14D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C22wrC2, C2xD12, S3xD4, D4.10D4, D6:D4, Q8.14D12

Character table of Q8.14D12

 class 12A2B2C2D34A4B4C4D4E4F4G4H4I6A6B6C8A8B12A12B12C12D12E12F12G12H24A24B
 size 112412222444121212242488242244444888
ρ1111111111111111111111111111111    trivial
ρ2111-1-1111-1111-11111-1-1-11111111-1-1-1    linear of order 2
ρ3111111111-1-1111-1111-1-111-11-1-1-11-1-1    linear of order 2
ρ4111-1-1111-1-1-11-11-111-11111-11-1-1-1-111    linear of order 2
ρ5111-11111-111-11-1-111-1-111111111-1-1-1    linear of order 2
ρ61111-1111111-1-1-1-11111-11111111111    linear of order 2
ρ7111-11111-1-1-1-11-1111-11-111-11-1-1-1-111    linear of order 2
ρ81111-11111-1-1-1-1-11111-1111-11-1-1-11-1-1    linear of order 2
ρ9222-20-122-2220000-1-11-20-1-1-1-1-1-1-1111    orthogonal lifted from D6
ρ1022-2022-220000-2002-2000220-2000000    orthogonal lifted from D4
ρ11222002-2-2000-202022000-2-20-2000000    orthogonal lifted from D4
ρ12222-20-122-2-2-20000-1-1120-1-11-11111-1-1    orthogonal lifted from D6
ρ1322220-1222220000-1-1-120-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ14222002-2-200020-2022000-2-20-2000000    orthogonal lifted from D4
ρ1522-2-2022-220000002-2-200-2-202000200    orthogonal lifted from D4
ρ1622220-1222-2-20000-1-1-1-20-1-11-1111-111    orthogonal lifted from D6
ρ1722-22022-2-20000002-2200-2-202000-200    orthogonal lifted from D4
ρ1822-20-22-2200002002-2000220-2000000    orthogonal lifted from D4
ρ1922-220-12-2-2000000-11-10011-√3-1√3-√3√31-√3√3    orthogonal lifted from D12
ρ2022-2-20-12-22000000-1110011√3-1-√3√3-√3-1-√3√3    orthogonal lifted from D12
ρ2122-220-12-2-2000000-11-10011√3-1-√3√3-√31√3-√3    orthogonal lifted from D12
ρ2222-2-20-12-22000000-1110011-√3-1√3-√3√3-1√3-√3    orthogonal lifted from D12
ρ2344400-2-4-40000000-2-20002202000000    orthogonal lifted from S3xD4
ρ2444-400-2-440000000-22000-2-202000000    orthogonal lifted from S3xD4
ρ254-400040002-20000-4000000-2022-2000    symplectic lifted from D4.10D4, Schur index 2
ρ264-40004000-220000-400000020-2-22000    symplectic lifted from D4.10D4, Schur index 2
ρ274-4000-2000-22000020000-2√32√3-1+√301+√31-√3-1-√3000    symplectic faithful, Schur index 2
ρ284-4000-20002-2000020000-2√32√31-√30-1-√3-1+√31+√3000    symplectic faithful, Schur index 2
ρ294-4000-20002-20000200002√3-2√31+√30-1+√3-1-√31-√3000    symplectic faithful, Schur index 2
ρ304-4000-2000-220000200002√3-2√3-1-√301-√31+√3-1+√3000    symplectic faithful, Schur index 2

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

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

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

Matrix representation of Q8.14D12 ►in GL4(F73) generated by

665900
14700
6659714
1475966
,
10710
01071
10720
01072
,
727222
10710
69311
7066720
,
83400
266500
6871820
125255
G:=sub<GL(4,GF(73))| [66,14,66,14,59,7,59,7,0,0,7,59,0,0,14,66],[1,0,1,0,0,1,0,1,71,0,72,0,0,71,0,72],[72,1,69,70,72,0,3,66,2,71,1,72,2,0,1,0],[8,26,68,12,34,65,7,5,0,0,18,2,0,0,20,55] >;
 

Q8.14D12 in GAP, Magma, Sage, TeX 

Q_8._{14}D_{12}
 
% in TeX
 
G:=Group("Q8.14D12");
 
// GroupNames label
 
G:=SmallGroup(192,385);
 
// by ID
 
G=gap.SmallGroup(192,385);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,58,1123,136,851,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=1,b^2=c^12=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a*b,d*c*d^-1=c^11>;
 
// generators/relations
 

Export

Character table of Q8.14D12 in TeX

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