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G = D4×Dic3order 96 = 25·3

Direct product of D4 and Dic3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4×Dic3, C23.22D6, C35(C4×D4), C123(C2×C4), (C3×D4)⋊3C4, C2.5(S3×D4), (C6×D4).4C2, (C2×D4).7S3, C41(C2×Dic3), (C2×C4).49D6, C6.37(C2×D4), C4⋊Dic313C2, (C4×Dic3)⋊4C2, C6.28(C4○D4), C6.D47C2, (C2×C6).49C23, C6.25(C22×C4), C222(C2×Dic3), C2.5(D42S3), (C2×C12).32C22, (C22×Dic3)⋊4C2, C2.6(C22×Dic3), C22.25(C22×S3), (C22×C6).17C22, (C2×Dic3).37C22, (C2×C6)⋊3(C2×C4), (C2×D4)(C2×Dic3), SmallGroup(96,141)

Series: Derived Chief Lower central Upper central

C1C6 — D4×Dic3
C1C3C6C2×C6C2×Dic3C22×Dic3 — D4×Dic3
C3C6 — D4×Dic3
C1C22C2×D4

Generators and relations for D4×Dic3
 G = < a,b,c,d | a4=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 178 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×4], C22×C6 [×2], C4×D4, C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, D4×Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, C2×Dic3 [×6], C22×S3, C4×D4, S3×D4, D42S3, C22×Dic3, D4×Dic3

Character table of D4×Dic3

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E6F6G12A12B
 size 111122222223333666666222444444
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-1-111111111-1-1-11-1111-1-1-1-111    linear of order 2
ρ31111-1-1-1-1111-1-1-1-1-1111-11111-1-1-1-111    linear of order 2
ρ411111111111-1-1-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ511111-1-111-1-1-1-1-1-111-1-111111-1-111-1-1    linear of order 2
ρ61111-111-11-1-1-1-1-1-11-1111-111111-1-1-1-1    linear of order 2
ρ71111-111-11-1-11111-11-1-1-1111111-1-1-1-1    linear of order 2
ρ811111-1-111-1-11111-1-111-1-1111-1-111-1-1    linear of order 2
ρ911-1-11-11-111-1-ii-ii-i-ii-iii-1-11-111-1-11    linear of order 4
ρ1011-1-1-11-1111-1-ii-ii-ii-iii-i-1-111-1-11-11    linear of order 4
ρ1111-1-11-11-111-1i-ii-iii-ii-i-i-1-11-111-1-11    linear of order 4
ρ1211-1-1-11-1111-1i-ii-ii-ii-i-ii-1-111-1-11-11    linear of order 4
ρ1311-1-1-1-1111-11i-ii-i-iii-ii-i-1-11-11-111-1    linear of order 4
ρ1411-1-111-1-11-11i-ii-i-i-i-iiii-1-111-11-11-1    linear of order 4
ρ1511-1-1-1-1111-11-ii-iii-i-ii-ii-1-11-11-111-1    linear of order 4
ρ1611-1-111-1-11-11-ii-iiiii-i-i-i-1-111-11-11-1    linear of order 4
ρ1722222222-1220000000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ182-22-20000200-2-222000000-22-2000000    orthogonal lifted from D4
ρ1922222-2-22-1-2-20000000000-1-1-111-1-111    orthogonal lifted from D6
ρ202222-222-2-1-2-20000000000-1-1-1-1-11111    orthogonal lifted from D6
ρ212-22-2000020022-2-2000000-22-2000000    orthogonal lifted from D4
ρ222222-2-2-2-2-1220000000000-1-1-11111-1-1    orthogonal lifted from D6
ρ2322-2-222-2-2-1-22000000000011-1-11-11-11    symplectic lifted from Dic3, Schur index 2
ρ2422-2-22-22-2-12-2000000000011-11-1-111-1    symplectic lifted from Dic3, Schur index 2
ρ2522-2-2-22-22-12-2000000000011-1-111-11-1    symplectic lifted from Dic3, Schur index 2
ρ2622-2-2-2-222-1-22000000000011-11-11-1-11    symplectic lifted from Dic3, Schur index 2
ρ272-2-220000200-2i2i2i-2i0000002-2-2000000    complex lifted from C4○D4
ρ282-2-2200002002i-2i-2i2i0000002-2-2000000    complex lifted from C4○D4
ρ294-44-40000-20000000000002-22000000    orthogonal lifted from S3×D4
ρ304-4-440000-2000000000000-222000000    symplectic lifted from D42S3, Schur index 2

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

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

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

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

D4×Dic3 is a maximal subgroup of
Dic34D8  D4.S3⋊C4  Dic36SD16  Dic3.D8  D4⋊Dic6  D4.Dic6  D4.2Dic6  D4⋊S3⋊C4  Dic3⋊D8  D8⋊Dic3  (C6×D8).C2  Dic33SD16  SD16⋊Dic3  (C3×D4).D4  C4×D42S3  D45Dic6  D46Dic6  C4×S3×D4  C4213D6  C42.108D6  C24.67D6  C24.43D6  C24.44D6  C24.46D6  C12⋊(C4○D4)  Dic619D4  C4⋊C4.178D6  C6.342+ 1+4  C6.702- 1+4  C6.712- 1+4  C4⋊C421D6  C6.732- 1+4  C6.432+ 1+4  C6.452+ 1+4  C6.462+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C4⋊C4.197D6  C6.802- 1+4  C6.1222+ 1+4  C6.852- 1+4  C42.139D6  C42.234D6  C42.143D6  C42.144D6  C42.166D6  C42.238D6  Dic611D4  C42.168D6  C24.49D6  C24.53D6  C6.1042- 1+4  C6.1442+ 1+4  C6.1452+ 1+4  D12⋊Dic3  C62.115C23  D208Dic3  Dic1516D4
D4×Dic3 is a maximal quotient of
C24.58D6  C24.19D6  C4⋊C45Dic3  C4⋊C46Dic3  C42.47D6  C123M4(2)  D8⋊Dic3  SD16⋊Dic3  Q16⋊Dic3  D85Dic3  D84Dic3  C24.29D6  C24.30D6  D12⋊Dic3  C62.115C23  D208Dic3  Dic1516D4

Matrix representation of D4×Dic3 in GL5(𝔽13)

10000
00100
012000
00010
00001
,
120000
00100
01000
00010
00001
,
120000
012000
001200
000121
000120
,
50000
05000
00500
00010
000112

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,12,0,0,0,1,0],[5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,1,1,0,0,0,0,12] >;

D4×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("D4xDic3");
// GroupNames label

G:=SmallGroup(96,141);
// by ID

G=gap.SmallGroup(96,141);
# by ID

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

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

Export

Character table of D4×Dic3 in TeX

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