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G = D6.D4order 96 = 25·3

2nd non-split extension by D6 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6.5D4, C4⋊C42S3, D6⋊C413C2, C2.12(S3×D4), (C2×C4).11D6, C6.25(C2×D4), Dic3⋊C46C2, (C2×D12).3C2, C6.12(C4○D4), (C2×C6).35C23, C2.14(C4○D12), (C2×C12).57C22, C2.5(Q83S3), C33(C22.D4), (C22×S3).6C22, C22.49(C22×S3), (C2×Dic3).11C22, (C3×C4⋊C4)⋊5C2, (S3×C2×C4)⋊13C2, SmallGroup(96,101)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D6.D4
C1C3C6C2×C6C22×S3S3×C2×C4 — D6.D4
C3C2×C6 — D6.D4
C1C22C4⋊C4

Generators and relations for D6.D4
 G = < a,b,c,d | a6=b2=c4=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 202 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], S3 [×3], C6 [×3], C2×C4 [×3], C2×C4 [×4], D4 [×2], C23 [×2], Dic3 [×2], C12 [×3], D6 [×2], D6 [×5], C2×C6, C22⋊C4 [×3], C4⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C22.D4, Dic3⋊C4, D6⋊C4 [×3], C3×C4⋊C4, S3×C2×C4, C2×D12, D6.D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C22×S3, C22.D4, C4○D12, S3×D4, Q83S3, D6.D4

Character table of D6.D4

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G6A6B6C12A12B12C12D12E12F
 size 11116612222446612222444444
ρ1111111111111111111111111    trivial
ρ2111111-1111-1-111-11111-11-1-1-1    linear of order 2
ρ311111111-1-1-11-1-1-1111-1-1-1-111    linear of order 2
ρ4111111-11-1-11-1-1-11111-11-11-1-1    linear of order 2
ρ51111-1-1-11-1-1-11111111-1-1-1-111    linear of order 2
ρ61111-1-111-1-11-111-1111-11-11-1-1    linear of order 2
ρ71111-1-1-111111-1-1-1111111111    linear of order 2
ρ81111-1-11111-1-1-1-111111-11-1-1-1    linear of order 2
ρ92222000-1-2-22-2000-1-1-11-11-111    orthogonal lifted from D6
ρ102222000-1-2-2-22000-1-1-11111-1-1    orthogonal lifted from D6
ρ112222000-12222000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ122-22-22-2020000000-22-2000000    orthogonal lifted from D4
ρ132222000-122-2-2000-1-1-1-11-1111    orthogonal lifted from D6
ρ142-22-2-22020000000-22-2000000    orthogonal lifted from D4
ρ1522-2-200020000-2i2i02-2-2000000    complex lifted from C4○D4
ρ1622-2-2000200002i-2i02-2-2000000    complex lifted from C4○D4
ρ172-2-2200022i-2i00000-2-222i0-2i000    complex lifted from C4○D4
ρ182-2-220002-2i2i00000-2-22-2i02i000    complex lifted from C4○D4
ρ192-2-22000-12i-2i0000011-1-i--3i-3-33    complex lifted from C4○D12
ρ202-2-22000-1-2i2i0000011-1i--3-i-33-3    complex lifted from C4○D12
ρ212-2-22000-1-2i2i0000011-1i-3-i--3-33    complex lifted from C4○D12
ρ222-2-22000-12i-2i0000011-1-i-3i--33-3    complex lifted from C4○D12
ρ2344-4-4000-20000000-222000000    orthogonal lifted from Q83S3, Schur index 2
ρ244-44-4000-200000002-22000000    orthogonal lifted from S3×D4

Smallest permutation representation of D6.D4
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 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 45)(14 44)(15 43)(16 48)(17 47)(18 46)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)
(1 29 17 24)(2 30 18 19)(3 25 13 20)(4 26 14 21)(5 27 15 22)(6 28 16 23)(7 34 46 39)(8 35 47 40)(9 36 48 41)(10 31 43 42)(11 32 44 37)(12 33 45 38)
(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 47 16 44)(14 48 17 45)(15 43 18 46)(19 34 22 31)(20 35 23 32)(21 36 24 33)(25 40 28 37)(26 41 29 38)(27 42 30 39)

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

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

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

D6.D4 is a maximal subgroup of
C6.2- 1+4  C6.112+ 1+4  C6.62- 1+4  C4212D6  C42.93D6  C42.100D6  C42.104D6  C4214D6  D1223D4  C4219D6  C42.131D6  C42.132D6  C42.133D6  C42.136D6  C6.372+ 1+4  C6.402+ 1+4  C6.442+ 1+4  C6.482+ 1+4  C4⋊C426D6  C6.162- 1+4  D1221D4  D1222D4  C6.532+ 1+4  C6.202- 1+4  C6.222- 1+4  C6.562+ 1+4  C6.782- 1+4  C6.592+ 1+4  S3×C22.D4  C6.1202+ 1+4  C6.1212+ 1+4  C6.822- 1+4  C6.612+ 1+4  C6.662+ 1+4  C6.672+ 1+4  C6.692+ 1+4  C42.237D6  C42.150D6  C42.151D6  C42.152D6  C42.153D6  C42.155D6  C42.157D6  C42.158D6  C4225D6  C4226D6  C42.189D6  C42.161D6  C42.163D6  C42.164D6  C4227D6  C42.171D6  D1212D4  C42.178D6  C42.180D6  D18.D4  C62.20C23  C62.23C23  C62.24C23  C62.54C23  D6.D12  C62.67C23  C62.238C23  D6⋊Dic5⋊C2  D30.35D4  (C2×D12).D5  D6.D20  D30.6D4  D30.7D4  D30.29D4
D6.D4 is a maximal quotient of
Dic3⋊C4⋊C4  (C2×C4).Dic6  C22.58(S3×D4)  (C2×C4)⋊9D12  D6⋊C45C4  C6.C22≀C2  C6.(C4⋊D4)  D6.2SD16  D6.4SD16  C4.Q8⋊S3  C6.(C4○D8)  D6.5D8  D6.2Q16  C2.D8⋊S3  C2.D87S3  C4⋊C45Dic3  (C2×C12).54D4  D6⋊C46C4  D6⋊C47C4  (C2×C4)⋊3D12  (C2×C12).289D4  (C2×C12).290D4  D18.D4  C62.20C23  C62.23C23  C62.24C23  C62.54C23  D6.D12  C62.67C23  C62.238C23  D6⋊Dic5⋊C2  D30.35D4  (C2×D12).D5  D6.D20  D30.6D4  D30.7D4  D30.29D4

Matrix representation of D6.D4 in GL4(𝔽13) generated by

0100
12100
0010
0001
,
3300
61000
00120
00012
,
11400
9200
0001
00120
,
3700
61000
00120
0001
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[3,6,0,0,3,10,0,0,0,0,12,0,0,0,0,12],[11,9,0,0,4,2,0,0,0,0,0,12,0,0,1,0],[3,6,0,0,7,10,0,0,0,0,12,0,0,0,0,1] >;

D6.D4 in GAP, Magma, Sage, TeX

D_6.D_4
% in TeX

G:=Group("D6.D4");
// GroupNames label

G:=SmallGroup(96,101);
// by ID

G=gap.SmallGroup(96,101);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,218,188,86,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^4=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D6.D4 in TeX

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