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G = D12.4D4  order 192 = 26·3

4th non-split extension by D12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.4D4, Dic6.4D4, M4(2).3D6, (C2xC12).8D4, (C2xC4).6D12, C4.80(S3xD4), C8.D6:7C2, C12.97(C2xD4), (C2xQ8).29D6, D12:C4:3C2, C6.17C22wrC2, C4.10D4:1S3, (C2xC12).9C23, Dic3:Q8:1C2, (C6xQ8).7C22, C3:1(D4.10D4), C4oD12.5C22, C22.12(C2xD12), C2.20(D6:D4), Q8.15D6.1C2, (C4xDic3).1C22, (C2xDic6).47C22, (C3xM4(2)).2C22, (C2xC6).22(C2xD4), (C2xC4).9(C22xS3), (C3xC4.10D4):3C2, SmallGroup(192,311)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — D12.4D4
Chief series
C1 — C3 — C6 — C12 — C2xC12 — C4oD12 — Q8.15D6 — D12.4D4
Lower central
C3 — C6 — C2xC12 — D12.4D4
Upper central
C1 — C2 — C2xC4 — C4.10D4

Generators and relations for D12.4D4
 G = < a,b,c,d | a12=b2=1, c4=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, dbd-1=a9b, dcd-1=a3c3 >

Subgroups: 432 in 142 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, Q8, Dic3, C12, C12, D6, C2xC6, C42, C4:C4, M4(2), SD16, Q16, C2xQ8, C2xQ8, C4oD4, C24, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xQ8, C4.10D4, C4wrC2, C4:Q8, C8.C22, 2- 1+4, C24:C2, Dic12, C4xDic3, Dic3:C4, C3xM4(2), C2xDic6, C4oD12, C4oD12, S3xQ8, Q8:3S3, C6xQ8, D4.10D4, D12:C4, C3xC4.10D4, C8.D6, Dic3:Q8, Q8.15D6, D12.4D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C22wrC2, C2xD12, S3xD4, D4.10D4, D6:D4, D12.4D4

Character table of D12.4D4

 class 12A2B2C2D34A4B4C4D4E4F4G4H4I6A6B8A8B12A12B12C12D24A24B24C24D
 size 1121212222441212121224248844888888
ρ1111111111111111111111111111    trivial
ρ2111-11111-1-111-11-111-1111-1-11-1-11    linear of order 2
ρ3111-1-111111-1-1-1-1-1111111111111    linear of order 2
ρ41111-1111-1-1-1-11-1111-1111-1-11-1-11    linear of order 2
ρ51111-1111-1-11-111-1111-111-1-1-111-1    linear of order 2
ρ6111-1-1111111-1-11111-1-11111-1-1-1-1    linear of order 2
ρ7111-11111-1-1-11-1-11111-111-1-1-111-1    linear of order 2
ρ81111111111-111-1-111-1-11111-1-1-1-1    linear of order 2
ρ922200-122-2-200000-1-1-22-1-111-111-1    orthogonal lifted from D6
ρ1022200-1222200000-1-122-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ11222002-2-2-22000002200-2-2-220000    orthogonal lifted from D4
ρ1222-20-22-2200020002-200-22000000    orthogonal lifted from D4
ρ13222002-2-22-2000002200-2-22-20000    orthogonal lifted from D4
ρ1422-2022-22000-20002-200-22000000    orthogonal lifted from D4
ρ1522200-122-2-200000-1-12-2-1-1111-1-11    orthogonal lifted from D6
ρ1622200-1222200000-1-1-2-2-1-1-1-11111    orthogonal lifted from D6
ρ1722-2-2022-200002002-2002-2000000    orthogonal lifted from D4
ρ1822-22022-20000-2002-2002-2000000    orthogonal lifted from D4
ρ1922200-1-2-22-200000-1-10011-11√3-√3√3-√3    orthogonal lifted from D12
ρ2022200-1-2-22-200000-1-10011-11-√3√3-√3√3    orthogonal lifted from D12
ρ2122200-1-2-2-2200000-1-100111-1-√3-√3√3√3    orthogonal lifted from D12
ρ2222200-1-2-2-2200000-1-100111-1√3√3-√3-√3    orthogonal lifted from D12
ρ2344-400-2-440000000-22002-2000000    orthogonal lifted from S3xD4
ρ2444-400-24-40000000-2200-22000000    orthogonal lifted from S3xD4
ρ254-400040000200-20-400000000000    symplectic lifted from D4.10D4, Schur index 2
ρ264-400040000-20020-400000000000    symplectic lifted from D4.10D4, Schur index 2
ρ278-8000-4000000000400000000000    symplectic faithful, Schur index 2

Smallest permutation representation of D12.4D4
►On 48 points
Generators in S48
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)

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

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

Matrix representation of D12.4D4 ►in GL8(F73)

10100000
01010000
720000000
072000000
000017100
000017200
000000722
000000721
,
660700000
066070000
140700000
014070000
000000171
000000172
000072200
000072100
,
0660590000
701400000
014070000
5906600000
00000010
00000001
000017100
000017200
,
6605900000
070140000
140700000
0590660000
0000005117
0000002322
0000682700
000045500

G:=sub<GL(8,GF(73))| [1,0,72,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1],[66,0,14,0,0,0,0,0,0,66,0,14,0,0,0,0,7,0,7,0,0,0,0,0,0,7,0,7,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0],[0,7,0,59,0,0,0,0,66,0,14,0,0,0,0,0,0,14,0,66,0,0,0,0,59,0,7,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[66,0,14,0,0,0,0,0,0,7,0,59,0,0,0,0,59,0,7,0,0,0,0,0,0,14,0,66,0,0,0,0,0,0,0,0,0,0,68,45,0,0,0,0,0,0,27,5,0,0,0,0,51,23,0,0,0,0,0,0,17,22,0,0] >;
 

D12.4D4 in GAP, Magma, Sage, TeX 

D_{12}._4D_4
 
% in TeX
 
G:=Group("D12.4D4");
 
// GroupNames label
 
G:=SmallGroup(192,311);
 
// by ID
 
G=gap.SmallGroup(192,311);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,58,1123,570,136,1684,438,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=a^3*c^3>;
 
// generators/relations
 

Export

Character table of D12.4D4 in TeX

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