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G = Dic3.D4order 96 = 25·3

1st non-split extension by Dic3 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3.6D4, C222Dic6, C23.17D6, (C2×C6)⋊Q8, C2.6(S3×D4), (C2×C4).5D6, C6.4(C2×Q8), C4⋊Dic32C2, C6.16(C2×D4), C31(C22⋊Q8), Dic3⋊C44C2, (C2×Dic6)⋊2C2, C22⋊C4.1S3, C2.6(C2×Dic6), C6.21(C4○D4), (C2×C12).1C22, (C2×C6).19C23, C2.6(D42S3), C6.D4.2C2, (C22×C6).8C22, C22.39(C22×S3), (C2×Dic3).5C22, (C22×Dic3).3C2, (C3×C22⋊C4).1C2, SmallGroup(96,85)

Series: Derived Chief Lower central Upper central

C1C2×C6 — Dic3.D4
C1C3C6C2×C6C2×Dic3C22×Dic3 — 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=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=a3c-1 >

Subgroups: 154 in 74 conjugacy classes, 35 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×2], C22×C6, C22⋊Q8, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, Dic3.D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C22×S3, C22⋊Q8, C2×Dic6, S3×D4, D42S3, Dic3.D4

Character table of Dic3.D4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C6D6E12A12B12C12D
 size 11112224466661212222444444
ρ1111111111111111111111111    trivial
ρ21111-1-11-111-11-1-11111-1-111-1-1    linear of order 2
ρ31111-1-111-11-11-11-1111-1-1-1-111    linear of order 2
ρ41111111-1-11111-1-111111-1-1-1-1    linear of order 2
ρ51111-1-11-11-11-111-1111-1-111-1-1    linear of order 2
ρ6111111111-1-1-1-1-1-1111111111    linear of order 2
ρ71111111-1-1-1-1-1-11111111-1-1-1-1    linear of order 2
ρ81111-1-111-1-11-11-11111-1-1-1-111    linear of order 2
ρ9222222-1-2-2000000-1-1-1-1-11111    orthogonal lifted from D6
ρ102-22-200200020-2002-2-2000000    orthogonal lifted from D4
ρ112222-2-2-12-2000000-1-1-11111-1-1    orthogonal lifted from D6
ρ12222222-122000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-22-2002000-202002-2-2000000    orthogonal lifted from D4
ρ142222-2-2-1-22000000-1-1-111-1-111    orthogonal lifted from D6
ρ1522-2-22-2200000000-22-2-220000    symplectic lifted from Q8, Schur index 2
ρ1622-2-2-22200000000-22-22-20000    symplectic lifted from Q8, Schur index 2
ρ1722-2-2-22-1000000001-11-11-333-3    symplectic lifted from Dic6, Schur index 2
ρ1822-2-22-2-1000000001-111-1-33-33    symplectic lifted from Dic6, Schur index 2
ρ1922-2-22-2-1000000001-111-13-33-3    symplectic lifted from Dic6, Schur index 2
ρ2022-2-2-22-1000000001-11-113-3-33    symplectic lifted from Dic6, Schur index 2
ρ212-2-22002002i0-2i000-2-22000000    complex lifted from C4○D4
ρ222-2-2200200-2i02i000-2-22000000    complex lifted from C4○D4
ρ234-44-400-200000000-222000000    orthogonal lifted from S3×D4
ρ244-4-4400-20000000022-2000000    symplectic lifted from D42S3, Schur index 2

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

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

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

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

Dic3.D4 is a maximal subgroup of
C233Dic6  C24.38D6  C24.42D6  C42.88D6  C42.90D6  C4212D6  C42.96D6  D4×Dic6  C42.102D6  D45Dic6  D46Dic6  C4214D6  C4219D6  C42.118D6  C24.67D6  C24.43D6  C24.44D6  C24.46D6  C6.322+ 1+4  Dic619D4  C6.702- 1+4  C6.712- 1+4  C6.722- 1+4  C6.732- 1+4  C6.492+ 1+4  (Q8×Dic3)⋊C2  C6.752- 1+4  S3×C22⋊Q8  Dic621D4  C6.512+ 1+4  C6.1182+ 1+4  C6.522+ 1+4  C6.792- 1+4  C4⋊C4.197D6  C6.802- 1+4  C6.812- 1+4  C6.822- 1+4  C6.1222+ 1+4  C6.632+ 1+4  C6.652+ 1+4  C6.852- 1+4  C6.692+ 1+4  C42.137D6  C42.139D6  C42.140D6  C42.141D6  Dic610D4  C42.144D6  C42.145D6  C42.159D6  C42.160D6  C42.161D6  C42.162D6  C42.164D6  C42.165D6  C222Dic18  D61Dic6  D62Dic6  D63Dic6  D64Dic6  C623Q8  C624Q8  C626Q8  Dic3.S4  D101Dic6  D102Dic6  Dic15.D4  D104Dic6  (C2×C10)⋊8Dic6  Dic15.48D4  C222Dic30
Dic3.D4 is a maximal quotient of
(C2×C12)⋊Q8  C6.(C4×Q8)  C2.(C4×Dic6)  Dic3⋊C4⋊C4  (C2×C4)⋊Dic6  C6.(C4⋊Q8)  (C2×C4).Dic6  (C22×C4).85D6  Dic3.D8  D4⋊Dic6  D4.Dic6  D4.2Dic6  Q82Dic6  Q83Dic6  Q8.3Dic6  Q8.4Dic6  C24.55D6  C24.57D6  C232Dic6  C24.17D6  C24.18D6  C24.58D6  C222Dic18  D61Dic6  D62Dic6  D63Dic6  D64Dic6  C623Q8  C624Q8  C626Q8  D101Dic6  D102Dic6  Dic15.D4  D104Dic6  (C2×C10)⋊8Dic6  Dic15.48D4  C222Dic30

Matrix representation of Dic3.D4 in GL4(𝔽13) generated by

1000
0100
00112
0010
,
12000
01200
0005
0050
,
01200
1000
0037
00610
,
1000
01200
0010
0001
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,0,5,0,0,5,0],[0,1,0,0,12,0,0,0,0,0,3,6,0,0,7,10],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;

Dic3.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_4
% in TeX

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

G:=SmallGroup(96,85);
// by ID

G=gap.SmallGroup(96,85);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,218,188,50,2309]);
// Polycyclic

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

Export

Character table of Dic3.D4 in TeX

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