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G = Dic34D4order 96 = 25·3

1st semidirect product of Dic3 and D4 acting through Inn(Dic3)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic34D4, C23.19D6, C3⋊D4⋊C4, C32(C4×D4), D62(C2×C4), C2.2(S3×D4), D6⋊C410C2, C22⋊C47S3, C222(C4×S3), C6.18(C2×D4), (C2×C4).28D6, Dic3⋊C49C2, Dic31(C2×C4), C6.7(C22×C4), (C4×Dic3)⋊11C2, C6.22(C4○D4), (C2×C6).22C23, C2.2(D42S3), (C2×C12).51C22, Dic32(C22⋊C4), (C22×Dic3)⋊1C2, C22.14(C22×S3), (C22×C6).11C22, (C22×S3).16C22, (C2×Dic3).47C22, (S3×C2×C4)⋊9C2, C2.9(S3×C2×C4), (C2×C6)⋊2(C2×C4), (C3×C22⋊C4)⋊9C2, (C2×C3⋊D4).2C2, C22⋊C4(C2×Dic3), SmallGroup(96,88)

Series: Derived Chief Lower central Upper central

C1C6 — Dic34D4
C1C3C6C2×C6C22×S3C2×C3⋊D4 — Dic34D4
C3C6 — Dic34D4
C1C22C22⋊C4

Generators and relations for Dic34D4
 G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 202 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, Dic3 [×4], Dic3, C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, C4×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, Dic34D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C22×S3, C4×D4, S3×C2×C4, S3×D4, D42S3, Dic34D4

Character table of Dic34D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E12A12B12C12D
 size 111122662222233336666222444444
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-1-11-11-111111-11-11111-1-1-1-111    linear of order 2
ρ3111111-1-11-1-1-1-111111-11-111111-1-1-1-1    linear of order 2
ρ41111-1-11111-11-11111-1-1-1-1111-1-111-1-1    linear of order 2
ρ5111111111-1-1-1-1-1-1-1-1-11-1111111-1-1-1-1    linear of order 2
ρ61111-1-1-1-111-11-1-1-1-1-11111111-1-111-1-1    linear of order 2
ρ7111111-1-111111-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ81111-1-1111-11-11-1-1-1-11-11-1111-1-1-1-111    linear of order 2
ρ911-1-11-11-11-iii-i-i-iii-i-1i1-1-111-1-iii-i    linear of order 4
ρ1011-1-1-11-111ii-i-i-i-iiii-1-i1-1-11-11i-ii-i    linear of order 4
ρ1111-1-1-111-11-i-iii-i-iiii1-i-1-1-11-11-ii-ii    linear of order 4
ρ1211-1-11-1-111i-i-ii-i-iii-i1i-1-1-111-1i-i-ii    linear of order 4
ρ1311-1-11-11-11i-i-iiii-i-ii-1-i1-1-111-1i-i-ii    linear of order 4
ρ1411-1-1-11-111-i-iiiii-i-i-i-1i1-1-11-11-ii-ii    linear of order 4
ρ1511-1-1-111-11ii-i-iii-i-i-i1i-1-1-11-11i-ii-i    linear of order 4
ρ1611-1-11-1-111-iii-iii-i-ii1-i-1-1-111-1-iii-i    linear of order 4
ρ172222-2-200-1-22-2200000000-1-1-11111-1-1    orthogonal lifted from D6
ρ182222-2-200-12-22-200000000-1-1-111-1-111    orthogonal lifted from D6
ρ192-22-2000020000-22-220000-22-2000000    orthogonal lifted from D4
ρ202-22-20000200002-22-20000-22-2000000    orthogonal lifted from D4
ρ2122222200-1222200000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ2222222200-1-2-2-2-200000000-1-1-1-1-11111    orthogonal lifted from D6
ρ2322-2-22-200-12i-2i-2i2i0000000011-1-11-iii-i    complex lifted from C4×S3
ρ2422-2-22-200-1-2i2i2i-2i0000000011-1-11i-i-ii    complex lifted from C4×S3
ρ2522-2-2-2200-12i2i-2i-2i0000000011-11-1-ii-ii    complex lifted from C4×S3
ρ2622-2-2-2200-1-2i-2i2i2i0000000011-11-1i-ii-i    complex lifted from C4×S3
ρ272-2-220000200002i-2i-2i2i00002-2-2000000    complex lifted from C4○D4
ρ282-2-22000020000-2i2i2i-2i00002-2-2000000    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 Dic34D4
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 24 4 21)(2 23 5 20)(3 22 6 19)(7 38 10 41)(8 37 11 40)(9 42 12 39)(13 25 16 28)(14 30 17 27)(15 29 18 26)(31 47 34 44)(32 46 35 43)(33 45 36 48)
(1 37 16 33)(2 42 17 32)(3 41 18 31)(4 40 13 36)(5 39 14 35)(6 38 15 34)(7 26 47 22)(8 25 48 21)(9 30 43 20)(10 29 44 19)(11 28 45 24)(12 27 46 23)
(1 36)(2 31)(3 32)(4 33)(5 34)(6 35)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)

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

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

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

Dic34D4 is a maximal subgroup of
C24.35D6  C24.38D6  C24.42D6  C42.188D6  C42.91D6  C4212D6  C42.96D6  C4×D42S3  C42.104D6  C4×S3×D4  C4213D6  C42.108D6  Dic623D4  C42.119D6  C24.67D6  C248D6  C24.44D6  C24.46D6  C12⋊(C4○D4)  Dic620D4  C6.342+ 1+4  C4⋊C421D6  C6.402+ 1+4  C6.422+ 1+4  C6.432+ 1+4  C6.442+ 1+4  C4⋊C4.187D6  C4⋊C426D6  Dic621D4  C6.1182+ 1+4  C6.522+ 1+4  C6.562+ 1+4  C6.782- 1+4  C4⋊C4.197D6  C6.1212+ 1+4  C6.822- 1+4  C4⋊C428D6  C6.1222+ 1+4  C6.632+ 1+4  C6.642+ 1+4  C6.662+ 1+4  C6.672+ 1+4  C6.852- 1+4  C42.233D6  C42.137D6  C42.138D6  Dic610D4  C4222D6  C4223D6  C42.234D6  C42.143D6  C42.160D6  C42.189D6  C42.161D6  C42.162D6  C42.163D6  C42.164D6  Dic94D4  C62.49C23  Dic34D12  C62.51C23  C62.72C23  C62.94C23  C62.115C23  C62.225C23  Dic32S4  Dic34D20  Dic1513D4  C1517(C4×D4)  Dic159D4  C1528(C4×D4)  Dic1517D4  Dic1519D4  C3⋊D4⋊F5
Dic34D4 is a maximal quotient of
C6.(C4×Q8)  Dic3⋊C42  C6.(C4×D4)  Dic3⋊C4⋊C4  D6⋊C42  D6⋊C4⋊C4  D6⋊C45C4  D6⋊C43C4  C3⋊D4⋊C8  D62M4(2)  Dic3⋊M4(2)  C3⋊C826D4  Dic34D8  D4.S3⋊C4  Dic36SD16  D4⋊S3⋊C4  Dic37SD16  C3⋊Q16⋊C4  Dic34Q16  Q83(C4×S3)  M4(2).22D6  C42.196D6  Dic3×C22⋊C4  C24.14D6  C24.15D6  C24.57D6  C24.23D6  C24.24D6  C24.60D6  Dic94D4  C62.49C23  Dic34D12  C62.51C23  C62.72C23  C62.94C23  C62.115C23  C62.225C23  Dic34D20  Dic1513D4  C1517(C4×D4)  Dic159D4  C1528(C4×D4)  Dic1517D4  Dic1519D4  C3⋊D4⋊F5

Matrix representation of Dic34D4 in GL4(𝔽13) generated by

11200
1000
00120
00012
,
0800
8000
0080
0008
,
0100
1000
0012
001212
,
12000
01200
001211
0001
G:=sub<GL(4,GF(13))| [1,1,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[0,8,0,0,8,0,0,0,0,0,8,0,0,0,0,8],[0,1,0,0,1,0,0,0,0,0,1,12,0,0,2,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,11,1] >;

Dic34D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_4D_4
% in TeX

G:=Group("Dic3:4D4");
// GroupNames label

G:=SmallGroup(96,88);
// by ID

G=gap.SmallGroup(96,88);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,188,50,2309]);
// Polycyclic

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

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Character table of Dic34D4 in TeX

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