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G = D6⋊3D4  order 96 = 25·3

3rd semidirect product of D6 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊3D4, C12⋊2D4, C23.13D6, (C6×D4)⋊3C2, (C2×D4)⋊4S3, C4⋊2(C3⋊D4), C3⋊4(C4⋊D4), (C2×C4).51D6, C6.50(C2×D4), C2.26(S3×D4), C4⋊Dic3⋊14C2, C6.31(C4○D4), (C2×C6).53C23, C6.D4⋊11C2, (C2×C12).34C22, C2.17(D4⋊2S3), (C22×C6).20C22, C22.60(C22×S3), (C22×S3).26C22, (C2×Dic3).19C22, (S3×C2×C4)⋊2C2, (C2×C3⋊D4)⋊5C2, C2.14(C2×C3⋊D4), SmallGroup(96,145)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D6⋊3D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — D6⋊3D4
Lower central
C3 — C2×C6 — D6⋊3D4
Upper central
C1 — C22 — C2×D4

Generators and relations for D6⋊3D4
 G = < a,b,c,d | a6=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >

Subgroups: 226 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, C4⋊Dic3, C6.D4, S3×C2×C4, C2×C3⋊D4, C6×D4, D6⋊3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D4⋊2S3, C2×C3⋊D4, D6⋊3D4

Character table of D6⋊3D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E6F6G12A12B
 size 11114466222661212222444444
ρ1111111111111111111111111    trivial
ρ21111-11111-1-1-1-11-1111-111-1-1-1    linear of order 2
ρ31111-11-1-11-1-111-11111-111-1-1-1    linear of order 2
ρ4111111-1-1111-1-1-1-1111111111    linear of order 2
ρ511111-1111-1-1-1-1-111111-1-11-1-1    linear of order 2
ρ61111-1-11111111-1-1111-1-1-1-111    linear of order 2
ρ71111-1-1-1-1111-1-111111-1-1-1-111    linear of order 2
ρ811111-1-1-11-1-1111-11111-1-11-1-1    linear of order 2
ρ92-22-2002-22000000-22-2000000    orthogonal lifted from D4
ρ102-2-2200002-220000-2-220000-22    orthogonal lifted from D4
ρ1122222-200-1-2-20000-1-1-1-111-111    orthogonal lifted from D6
ρ122-2-22000022-20000-2-2200002-2    orthogonal lifted from D4
ρ1322222200-1220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ142222-2-200-1220000-1-1-11111-1-1    orthogonal lifted from D6
ρ152222-2200-1-2-20000-1-1-11-1-1111    orthogonal lifted from D6
ρ162-22-200-222000000-22-2000000    orthogonal lifted from D4
ρ172-2-220000-1-22000011-1-√-3-√-3√-3√-31-1    complex lifted from C3⋊D4
ρ182-2-220000-1-22000011-1√-3√-3-√-3-√-31-1    complex lifted from C3⋊D4
ρ192-2-220000-12-2000011-1√-3-√-3√-3-√-3-11    complex lifted from C3⋊D4
ρ202-2-220000-12-2000011-1-√-3√-3-√-3√-3-11    complex lifted from C3⋊D4
ρ2122-2-200002002i-2i002-2-2000000    complex lifted from C4○D4
ρ2222-2-20000200-2i2i002-2-2000000    complex lifted from C4○D4
ρ234-44-40000-20000002-22000000    orthogonal lifted from S3×D4
ρ2444-4-40000-2000000-222000000    symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of D6⋊3D4
►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 6)(2 5)(3 4)(8 12)(9 11)(13 14)(15 18)(16 17)(19 21)(22 24)(25 29)(26 28)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)(43 45)(46 48)

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

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

D6⋊3D4 is a maximal subgroup of
 D6.D8  D6⋊D8  D6.SD16  D6⋊SD16  D6⋊C8⋊11C2  C3⋊C8⋊1D4  C3⋊C8⋊D4  C24⋊1C4⋊C2  D12⋊D4  D6⋊3D8  Dic6⋊D4  C24⋊12D4  D6⋊8SD16  C24⋊14D4  D12⋊7D4  C24⋊8D4  C42.228D6  D12⋊24D4  C42.229D6  C42.113D6  C42.115D6  C42.116D6  C42.117D6  C24⋊7D6  C24.44D6  C24.46D6  C24.47D6  S3×C4⋊D4  C4⋊C4⋊21D6  C6.382+ 1+4  C6.722- 1+4  C6.732- 1+4  D12⋊20D4  C6.422+ 1+4  C6.432+ 1+4  C6.442+ 1+4  C6.452+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.492+ 1+4  C6.612+ 1+4  C6.632+ 1+4  C6.642+ 1+4  C6.692+ 1+4  D12⋊10D4  Dic6⋊10D4  C42.234D6  C42.144D6  C42.238D6  D12⋊11D4  Dic6⋊11D4  C42.168D6  D4×C3⋊D4  C24.52D6  C24.53D6  (C2×D4)⋊43D6  C6.1072- 1+4  C6.1082- 1+4  C6.1482+ 1+4  C36⋊2D4  D6⋊2D12  C12⋊2D12  C62.100C23  C62.112C23  C62.256C23  C60⋊4D4  C12⋊2D20  D30⋊6D4  (S3×C10)⋊D4  C60⋊2D4
D6⋊3D4 is a maximal quotient of
 C24.14D6  C24.17D6  C24.23D6  C24.27D6  C12⋊(C4⋊C4)  (C2×Dic3).Q8  C4⋊(D6⋊C4)  (C2×C12).289D4  C42.61D6  D12.23D4  C12⋊2D8  Dic6⋊9D4  C12⋊5SD16  C12⋊Q16  D6⋊3D8  C24⋊12D4  C24.23D4  C24⋊14D4  C24⋊8D4  C24.44D4  D6⋊3Q16  C24.36D4  C24.29D4  C24.30D6  C24.31D6  C24.32D6  C36⋊2D4  D6⋊2D12  C12⋊2D12  C62.100C23  C62.112C23  C62.256C23  C60⋊4D4  C12⋊2D20  D30⋊6D4  (S3×C10)⋊D4  C60⋊2D4

Matrix representation of D6⋊3D4 ►in GL4(𝔽13) generated by

0100
12100
00120
00012
,
11200
01200
00120
0001
,
12000
01200
0050
0008
,
2900
41100
0008
0050
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,12,12,0,0,0,0,12,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,8],[2,4,0,0,9,11,0,0,0,0,0,5,0,0,8,0] >;
 

D6⋊3D4 in GAP, Magma, Sage, TeX 

D_6\rtimes_3D_4
 
% in TeX
 
G:=Group("D6:3D4");
 
// GroupNames label
 
G:=SmallGroup(96,145);
 
// by ID
 
G=gap.SmallGroup(96,145);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D6⋊3D4 in TeX

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