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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, (C2×C12).8D4, (C2×C4).6D12, C4.80(S3×D4), C8.D6⋊7C2, C12.97(C2×D4), (C2×Q8).29D6, D12⋊C4⋊3C2, C6.17C22≀C2, C4.10D4⋊1S3, (C2×C12).9C23, Dic3⋊Q8⋊1C2, (C6×Q8).7C22, C3⋊1(D4.10D4), C4○D12.5C22, C22.12(C2×D12), C2.20(D6⋊D4), Q8.15D6.1C2, (C4×Dic3).1C22, (C2×Dic6).47C22, (C3×M4(2)).2C22, (C2×C6).22(C2×D4), (C2×C4).9(C22×S3), (C3×C4.10D4)⋊3C2, SmallGroup(192,311)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — D12.4D4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8.15D6 — D12.4D4
Lower central
C3 — C6 — C2×C12 — D12.4D4
Upper central
C1 — C2 — C2×C4 — 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, C2×C4, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C24⋊C2, Dic12, C4×Dic3, Dic3⋊C4, C3×M4(2), C2×Dic6, C4○D12, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, D4.10D4, D12⋊C4, C3×C4.10D4, C8.D6, Dic3⋊Q8, Q8.15D6, D12.4D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, 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 S3×D4
ρ2444-400-24-40000000-2200-22000000    orthogonal lifted from S3×D4
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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