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G = D4⋊Dic3  order 96 = 25·3

1st semidirect product of D4 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.7D4, C6.12D8, D4⋊1Dic3, C6.6SD16, (C3×D4)⋊1C4, C12.7(C2×C4), (C2×D4).1S3, (C6×D4).1C2, (C2×C4).39D6, (C2×C6).33D4, C3⋊3(D4⋊C4), C4⋊Dic3⋊10C2, C2.3(D4⋊S3), C4.1(C2×Dic3), C4.12(C3⋊D4), C2.3(D4.S3), C6.13(C22⋊C4), (C2×C12).16C22, C2.3(C6.D4), C22.17(C3⋊D4), (C2×C3⋊C8)⋊2C2, SmallGroup(96,39)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D4⋊Dic3
Chief series
C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — D4⋊Dic3
Lower central
C3 — C6 — C12 — D4⋊Dic3
Upper central
C1 — C22 — C2×C4 — C2×D4

Generators and relations for D4⋊Dic3
 G = < a,b,c,d | a4=b2=c6=1, d2=c3, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

C1
C2
C2
C2
4C2
4C2
C3
C4
C22
C4
2C22
2C22
4C22
4C22
12C4
C6
C6
C6
4C6
4C6
D4
C2×C4
D4
2D4
2C23
6C2×C4
6C8
C12
C12
C2×C6
2C2×C6
2C2×C6
4C2×C6
4Dic3
4C2×C6
C2×D4
3C4⋊C4
3C2×C8
C2×C12
C3×D4
C3×D4
2C3⋊C8
2C3×D4
2C2×Dic3
2C22×C6
3D4⋊C4
C6×D4
C4⋊Dic3
C2×C3⋊C8
D4⋊Dic3

Character table of D4⋊Dic3

 class 12A2B2C2D2E34A4B4C4D6A6B6C6D6E6F6G8A8B8C8D12A12B
 size 11114422212122224444666644
ρ1111111111111111111111111    trivial
ρ2111111111-1-11111111-1-1-1-111    linear of order 2
ρ31111-1-111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ41111-1-1111-1-1111-1-1-1-1111111    linear of order 2
ρ511-1-1-111-11i-i1-1-111-1-1-ii-ii-11    linear of order 4
ρ611-1-1-111-11-ii1-1-111-1-1i-ii-i-11    linear of order 4
ρ711-1-11-11-11i-i1-1-1-1-111i-ii-i-11    linear of order 4
ρ811-1-11-11-11-ii1-1-1-1-111-ii-ii-11    linear of order 4
ρ92222002-2-20022200000000-2-2    orthogonal lifted from D4
ρ10222222-12200-1-1-1-1-1-1-10000-1-1    orthogonal lifted from S3
ρ112-22-20020000-2-220000√2√2-√2-√200    orthogonal lifted from D8
ρ122222-2-2-12200-1-1-111110000-1-1    orthogonal lifted from D6
ρ1322-2-20022-2002-2-2000000002-2    orthogonal lifted from D4
ρ142-22-20020000-2-220000-√2-√2√2√200    orthogonal lifted from D8
ρ1522-2-22-2-1-2200-11111-1-100001-1    symplectic lifted from Dic3, Schur index 2
ρ1622-2-2-22-1-2200-111-1-11100001-1    symplectic lifted from Dic3, Schur index 2
ρ1722-2-200-12-200-111√-3-√-3-√-3√-30000-11    complex lifted from C3⋊D4
ρ18222200-1-2-200-1-1-1-√-3√-3-√-3√-3000011    complex lifted from C3⋊D4
ρ1922-2-200-12-200-111-√-3√-3√-3-√-30000-11    complex lifted from C3⋊D4
ρ202-2-220020000-22-20000-√-2√-2√-2-√-200    complex lifted from SD16
ρ212-2-220020000-22-20000√-2-√-2-√-2√-200    complex lifted from SD16
ρ22222200-1-2-200-1-1-1√-3-√-3√-3-√-3000011    complex lifted from C3⋊D4
ρ234-44-400-2000022-20000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ244-4-4400-200002-220000000000    symplectic lifted from D4.S3, Schur index 2

Smallest permutation representation of D4⋊Dic3
►On 48 points
Generators in S48
(1 27 23 7)(2 28 24 8)(3 29 19 9)(4 30 20 10)(5 25 21 11)(6 26 22 12)(13 41 33 46)(14 42 34 47)(15 37 35 48)(16 38 36 43)(17 39 31 44)(18 40 32 45)

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)

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

D4⋊Dic3 is a maximal subgroup of
 Dic3.D8  Dic3.SD16  D4⋊Dic6  D4.Dic6  C4⋊C4.D6  C12⋊Q8⋊C2  D4.2Dic6  (C2×C8).200D6  S3×D4⋊C4  C4⋊C4⋊19D6  D4⋊(C4×S3)  D4⋊2S3⋊C4  D6.D8  D6.SD16  D6⋊C8⋊11C2  C24⋊1C4⋊C2  C12.50D8  C12.38SD16  D4.3Dic6  C4×D4⋊S3  C42.48D6  C4×D4.S3  C42.51D6  (C2×C6).D8  C4⋊D4.S3  C6.Q16⋊C2  C3⋊C8⋊22D4  C4⋊D4⋊S3  C3⋊C8⋊23D4  C3⋊C8⋊5D4  C42.61D6  C42.62D6  C42.213D6  D12.23D4  C12.16D8  C42.72D6  C12⋊2D8  Dic6⋊9D4  Dic3×D8  Dic3⋊D8  D8⋊Dic3  (C6×D8).C2  D12⋊D4  D6⋊3D8  Dic6⋊D4  C24⋊12D4  Dic3×SD16  Dic3⋊5SD16  SD16⋊Dic3  (C3×Q8).D4  D6⋊6SD16  C24⋊14D4  Dic6.16D4  C24⋊8D4  (C6×D4)⋊6C4  (C2×C6)⋊8D8  (C3×D4).31D4  C4○D4⋊3Dic3  C4○D4⋊4Dic3  (C3×D4)⋊14D4  (C3×D4).32D4  D4⋊Dic9  D12⋊3Dic3  C6.16D24  C62.116D4  C30.D8  C6.D40  D4⋊Dic15  D20⋊Dic3
D4⋊Dic3 is a maximal quotient of
 C12.C42  C12.57D8  (C6×D4)⋊C4  C12.9D8  C12.10D8  D8⋊1Dic3  D8.Dic3  C6.5Q32  Q16.Dic3  D8⋊2Dic3  C24.41D4  D4⋊Dic9  D12⋊3Dic3  C6.16D24  C62.116D4  C30.D8  C6.D40  D4⋊Dic15  D20⋊Dic3

Matrix representation of D4⋊Dic3 ►in GL4(𝔽73) generated by

1000
0100
00166
004272
,
1000
0100
00166
00072
,
65000
71900
0010
0001
,
272600
04600
00039
00150
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,42,0,0,66,72],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,66,72],[65,71,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,26,46,0,0,0,0,0,15,0,0,39,0] >;
 

D4⋊Dic3 in GAP, Magma, Sage, TeX 

D_4\rtimes {\rm Dic}_3
 
% in TeX
 
G:=Group("D4:Dic3");
 
// GroupNames label
 
G:=SmallGroup(96,39);
 
// by ID
 
G=gap.SmallGroup(96,39);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,579,297,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^6=1,d^2=c^3,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D4⋊Dic3 in TeX
Character table of D4⋊Dic3 in TeX

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