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G = Dic3⋊D4order 96 = 25·3

1st semidirect product of Dic3 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D61D4, Dic32D4, C23.10D6, (C2×C4).7D6, C2.9(S3×D4), D6⋊C411C2, (C2×D12)⋊3C2, C31(C4⋊D4), C22⋊C44S3, C6.20(C2×D4), Dic3⋊C45C2, C6.9(C4○D4), (C2×C6).25C23, C2.11(C4○D12), (C2×C12).53C22, (C22×C6).14C22, C22.43(C22×S3), (C2×Dic3).7C22, (C22×S3).18C22, (S3×C2×C4)⋊11C2, (C2×C3⋊D4)⋊2C2, (C3×C22⋊C4)⋊6C2, SmallGroup(96,91)

Series: Derived Chief Lower central Upper central

C1C2×C6 — Dic3⋊D4
C1C3C6C2×C6C22×S3S3×C2×C4 — Dic3⋊D4
C3C2×C6 — Dic3⋊D4
C1C22C22⋊C4

Generators and relations for Dic3⋊D4
 G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=cac-1=dad=a-1, cbc-1=a3b, bd=db, dcd=c-1 >

Subgroups: 250 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], S3 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], Dic3 [×2], Dic3, C12 [×2], D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], Dic3⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4 [×2], Dic3⋊D4

Character table of Dic3⋊D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E12A12B12C12D
 size 11114661222246612222444444
ρ1111111111111111111111111    trivial
ρ211111-1-111-1-1-111-111111-1-1-1-1    linear of order 2
ρ31111-1-1-11111-1-1-11111-1-111-1-1    linear of order 2
ρ41111-11111-1-11-1-1-1111-1-1-1-111    linear of order 2
ρ51111-1-1-1-11-1-11111111-1-1-1-111    linear of order 2
ρ61111-111-1111-111-1111-1-111-1-1    linear of order 2
ρ71111111-11-1-1-1-1-1111111-1-1-1-1    linear of order 2
ρ811111-1-1-11111-1-1-1111111111    linear of order 2
ρ922222000-1-2-2-2000-1-1-1-1-11111    orthogonal lifted from D6
ρ102-22-200002000-220-22-2000000    orthogonal lifted from D4
ρ112222-2000-1-2-22000-1-1-11111-1-1    orthogonal lifted from D6
ρ1222-2-20-22020000002-2-2000000    orthogonal lifted from D4
ρ1322-2-202-2020000002-2-2000000    orthogonal lifted from D4
ρ1422222000-1222000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ152222-2000-122-2000-1-1-111-1-111    orthogonal lifted from D6
ρ162-22-2000020002-20-22-2000000    orthogonal lifted from D4
ρ172-2-2200002-2i2i0000-2-2200-2i2i00    complex lifted from C4○D4
ρ182-2-22000022i-2i0000-2-22002i-2i00    complex lifted from C4○D4
ρ192-2-220000-1-2i2i000011-1-3--3i-i3-3    complex lifted from C4○D12
ρ202-2-220000-12i-2i000011-1--3-3-ii3-3    complex lifted from C4○D12
ρ212-2-220000-1-2i2i000011-1--3-3i-i-33    complex lifted from C4○D12
ρ222-2-220000-12i-2i000011-1-3--3-ii-33    complex lifted from C4○D12
ρ2344-4-40000-2000000-222000000    orthogonal lifted from S3×D4
ρ244-44-40000-20000002-22000000    orthogonal lifted from S3×D4

Smallest permutation representation of Dic3⋊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 28 4 25)(2 27 5 30)(3 26 6 29)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 21 16 24)(14 20 17 23)(15 19 18 22)(37 45 40 48)(38 44 41 47)(39 43 42 46)
(1 36 13 37)(2 35 14 42)(3 34 15 41)(4 33 16 40)(5 32 17 39)(6 31 18 38)(7 19 44 26)(8 24 45 25)(9 23 46 30)(10 22 47 29)(11 21 48 28)(12 20 43 27)
(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)

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

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

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

Dic3⋊D4 is a maximal subgroup of
C24.38D6  C24.41D6  C42.93D6  C42.95D6  C42.97D6  C42.100D6  C42.104D6  C4214D6  C42.228D6  D1223D4  Dic624D4  C42.113D6  C4219D6  C42.116D6  C247D6  C248D6  C24.45D6  C24.47D6  C6.322+ 1+4  Dic620D4  S3×C4⋊D4  C6.722- 1+4  D1219D4  C6.402+ 1+4  C6.442+ 1+4  C6.482+ 1+4  C6.172- 1+4  D1222D4  Dic622D4  C6.202- 1+4  C6.222- 1+4  C6.592+ 1+4  C6.1202+ 1+4  C6.1212+ 1+4  C6.822- 1+4  C4⋊C428D6  C6.612+ 1+4  C6.642+ 1+4  C6.662+ 1+4  C6.682+ 1+4  C6.692+ 1+4  C42.233D6  C42.138D6  C4220D6  D1210D4  Dic610D4  C4222D6  C42.145D6  C4225D6  C42.189D6  C42.161D6  C42.163D6  C42.164D6  C4227D6  D18⋊D4  C62.55C23  D6⋊D12  Dic33D12  C62.82C23  C62.100C23  C62.113C23  C62.228C23  Dic152D4  D6⋊D20  D302D4  D3012D4  D307D4  Dic154D4  D309D4
Dic3⋊D4 is a maximal quotient of
C6.(C4×Q8)  C6.(C4⋊Q8)  (C2×C4)⋊9D12  D6⋊(C4⋊C4)  D6⋊C45C4  C6.C22≀C2  C6.(C4⋊D4)  (C22×C4).37D6  D12.2D4  D12.3D4  D12.6D4  D12.7D4  Dic62D4  Dic6.D4  D6⋊D8  D6⋊SD16  C3⋊C81D4  C3⋊C8⋊D4  D123D4  D12.D4  Dic3⋊Q16  Dic6.11D4  D62SD16  C3⋊(C8⋊D4)  D61Q16  C3⋊C8.D4  Dic3⋊SD16  D12.12D4  C24.17D6  C24.19D6  C24.23D6  C24.24D6  C24.25D6  C233D12  D18⋊D4  C62.55C23  D6⋊D12  Dic33D12  C62.82C23  C62.100C23  C62.113C23  C62.228C23  Dic152D4  D6⋊D20  D302D4  D3012D4  D307D4  Dic154D4  D309D4

Matrix representation of Dic3⋊D4 in GL6(𝔽13)

010000
12120000
0012000
0001200
0000120
0000012
,
1200000
110000
0071000
008600
0000123
000081
,
100000
12120000
005000
006800
0000123
000001
,
1200000
110000
0091100
001400
000010
000001

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,12,8,0,0,0,0,3,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,5,6,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,3,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,11,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes D_4
% in TeX

G:=Group("Dic3:D4");
// GroupNames label

G:=SmallGroup(96,91);
// by ID

G=gap.SmallGroup(96,91);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,506,188,2309]);
// Polycyclic

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

Export

Character table of Dic3⋊D4 in TeX

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