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G = D4⋊Dic3order 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, D41Dic3, C6.6SD16, (C3×D4)⋊1C4, C12.7(C2×C4), (C2×D4).1S3, (C6×D4).1C2, (C2×C4).39D6, (C2×C6).33D4, C33(D4⋊C4), C4⋊Dic310C2, 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

C1C12 — D4⋊Dic3
C1C3C6C2×C6C2×C12C4⋊Dic3 — D4⋊Dic3
C3C6C12 — D4⋊Dic3
C1C22C2×C4C2×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 >

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

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
ρ112-22-20020000-2-22000022-2-200    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
ρ142-22-20020000-2-220000-2-22200    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⋊C419D6  D4⋊(C4×S3)  D42S3⋊C4  D6.D8  D6.SD16  D6⋊C811C2  C241C4⋊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⋊C822D4  C4⋊D4⋊S3  C3⋊C823D4  C3⋊C85D4  C42.61D6  C42.62D6  C42.213D6  D12.23D4  C12.16D8  C42.72D6  C122D8  Dic69D4  Dic3×D8  Dic3⋊D8  D8⋊Dic3  (C6×D8).C2  D12⋊D4  D63D8  Dic6⋊D4  C2412D4  Dic3×SD16  Dic35SD16  SD16⋊Dic3  (C3×Q8).D4  D66SD16  C2414D4  Dic6.16D4  C248D4  (C6×D4)⋊6C4  (C2×C6)⋊8D8  (C3×D4).31D4  C4○D43Dic3  C4○D44Dic3  (C3×D4)⋊14D4  (C3×D4).32D4  D4⋊Dic9  D123Dic3  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  D81Dic3  D8.Dic3  C6.5Q32  Q16.Dic3  D82Dic3  C24.41D4  D4⋊Dic9  D123Dic3  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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