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G = Dic35D4order 96 = 25·3

2nd semidirect product of Dic3 and D4 acting through Inn(Dic3)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D125C4, Dic35D4, C33(C4×D4), C4⋊C48S3, C41(C4×S3), C122(C2×C4), D63(C2×C4), C2.4(S3×D4), D6⋊C412C2, C6.24(C2×D4), (C2×C4).31D6, Dic33(C4⋊C4), (C4×Dic3)⋊3C2, (C2×D12).7C2, C6.33(C4○D4), (C2×C6).34C23, C6.11(C22×C4), (C2×C12).24C22, C2.2(Q83S3), C22.18(C22×S3), (C22×S3).20C22, (C2×Dic3).49C22, (C3×C4⋊C4)⋊4C2, (S3×C2×C4)⋊12C2, C2.13(S3×C2×C4), C4⋊C4(C2×Dic3), SmallGroup(96,100)

Series: Derived Chief Lower central Upper central

C1C6 — Dic35D4
C1C3C6C2×C6C22×S3C2×D12 — Dic35D4
C3C6 — Dic35D4
C1C22C4⋊C4

Generators and relations for Dic35D4
 G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 218 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], Dic3 [×2], Dic3, C12 [×2], C12 [×2], D6 [×4], D6 [×4], C2×C6, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C4×D4, C4×Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, Dic35D4
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, Q83S3, Dic35D4

Character table of Dic35D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C12A12B12C12D12E12F
 size 111166662222222333366222444444
ρ1111111111111111111111111111111    trivial
ρ211111-11-11-1-1-111-1-1-1-1-11111111-1-1-1-1    linear of order 2
ρ31111-1-1-1-11111111-1-1-1-1-1-1111111111    linear of order 2
ρ41111-11-111-1-1-111-11111-1-111111-1-1-1-1    linear of order 2
ρ51111111111-1-1-1-11-1-1-1-1-1-1111-1-1-11-11    linear of order 2
ρ611111-11-11-111-1-1-11111-1-1111-1-11-11-1    linear of order 2
ρ71111-1-1-1-111-1-1-1-11111111111-1-1-11-11    linear of order 2
ρ81111-11-111-111-1-1-1-1-1-1-111111-1-11-11-1    linear of order 2
ρ911-1-11-1-1111i-ii-i-1-i-iiii-i-1-11-ii-i-1i1    linear of order 4
ρ1011-1-111-1-11-1-iii-i1ii-i-ii-i-1-11-iii1-i-1    linear of order 4
ρ1111-1-1-111-111i-ii-i-1ii-i-i-ii-1-11-ii-i-1i1    linear of order 4
ρ1211-1-1-1-1111-1-iii-i1-i-iii-ii-1-11-iii1-i-1    linear of order 4
ρ1311-1-111-1-11-1i-i-ii1-i-iii-ii-1-11i-i-i1i-1    linear of order 4
ρ1411-1-11-1-1111-ii-ii-1ii-i-i-ii-1-11i-ii-1-i1    linear of order 4
ρ1511-1-1-1-1111-1i-i-ii1ii-i-ii-i-1-11i-i-i1i-1    linear of order 4
ρ1611-1-1-111-111-ii-ii-1-i-iiii-i-1-11i-ii-1-i1    linear of order 4
ρ1722220000-1-222-2-2-2000000-1-1-111-11-11    orthogonal lifted from D6
ρ1822220000-1-2-2-222-2000000-1-1-1-1-11111    orthogonal lifted from D6
ρ192-22-2000020000002-22-200-22-2000000    orthogonal lifted from D4
ρ2022220000-12-2-2-2-22000000-1-1-1111-11-1    orthogonal lifted from D6
ρ2122220000-1222222000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ222-22-200002000000-22-2200-22-2000000    orthogonal lifted from D4
ρ2322-2-20000-122i-2i2i-2i-200000011-1i-ii1-i-1    complex lifted from C4×S3
ρ2422-2-20000-12-2i2i-2i2i-200000011-1-ii-i1i-1    complex lifted from C4×S3
ρ252-2-22000020000002i-2i-2i2i002-2-2000000    complex lifted from C4○D4
ρ262-2-2200002000000-2i2i2i-2i002-2-2000000    complex lifted from C4○D4
ρ2722-2-20000-1-2-2i2i2i-2i200000011-1i-i-i-1i1    complex lifted from C4×S3
ρ2822-2-20000-1-22i-2i-2i2i200000011-1-iii-1-i1    complex lifted from C4×S3
ρ294-44-40000-20000000000002-22000000    orthogonal lifted from S3×D4
ρ304-4-440000-2000000000000-222000000    orthogonal lifted from Q83S3, Schur index 2

Smallest permutation representation of Dic35D4
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 11 4 8)(2 10 5 7)(3 9 6 12)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 33 22 36)(20 32 23 35)(21 31 24 34)(25 37 28 40)(26 42 29 39)(27 41 30 38)
(1 40 16 32)(2 41 17 33)(3 42 18 34)(4 37 13 35)(5 38 14 36)(6 39 15 31)(7 27 43 19)(8 28 44 20)(9 29 45 21)(10 30 46 22)(11 25 47 23)(12 26 48 24)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 45)(8 44)(9 43)(10 48)(11 47)(12 46)(19 21)(22 24)(26 30)(27 29)(31 33)(34 36)(38 42)(39 41)

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

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

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

Dic35D4 is a maximal subgroup of
Dic34D8  D4⋊S3⋊C4  D123D4  D12.D4  Dic37SD16  Q83(C4×S3)  Dic3⋊SD16  D12.12D4  Dic38SD16  D249C4  D12⋊Q8  D12.Q8  Dic35D8  C24⋊C2⋊C4  D122Q8  D12.2Q8  C6.82+ 1+4  C6.2- 1+4  C6.112+ 1+4  C429D6  C42.188D6  C42.95D6  C42.97D6  C4×S3×D4  C4213D6  Dic624D4  C42.114D6  C42.122D6  C4×Q83S3  C42.126D6  C42.136D6  C12⋊(C4○D4)  D1219D4  D1220D4  C6.472+ 1+4  C4⋊C4.187D6  C4⋊C426D6  D1221D4  D1222D4  Dic622D4  C6.532+ 1+4  C6.202- 1+4  C6.242- 1+4  C6.1212+ 1+4  C4⋊C428D6  C6.612+ 1+4  C6.642+ 1+4  D127Q8  C42.237D6  C42.150D6  C42.151D6  C42.153D6  C42.155D6  C4225D6  C4226D6  C42.189D6  C42.163D6  C42.240D6  D128Q8  D129Q8  C42.178D6  D36⋊C4  C62.51C23  Dic35D12  D12⋊Dic3  C62.74C23  C62.237C23  Dic1514D4  D6014C4  Dic158D4  C1522(C4×D4)  D6011C4  D603C4
Dic35D4 is a maximal quotient of
C2.(C4×Dic6)  (C2×C4)⋊9D12  D6⋊C42  D6⋊C45C4  D12⋊C8  D63M4(2)  C122M4(2)  Dic38SD16  Dic129C4  D249C4  Dic35D8  Dic35Q16  C24⋊C2⋊C4  D2410C4  D247C4  C12⋊(C4⋊C4)  Dic3×C4⋊C4  (C2×D12)⋊10C4  D6⋊C46C4  D6⋊C47C4  D36⋊C4  C62.51C23  Dic35D12  D12⋊Dic3  C62.74C23  C62.237C23  Dic1514D4  D6014C4  Dic158D4  C1522(C4×D4)  D6011C4  D603C4

Matrix representation of Dic35D4 in GL5(𝔽13)

120000
00100
0121200
00010
00001
,
80000
012000
01100
00010
00001
,
10000
01000
00100
000123
00081
,
120000
01000
0121200
000120
00081

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,8,0,0,0,3,1],[12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,12,8,0,0,0,0,1] >;

Dic35D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_5D_4
% in TeX

G:=Group("Dic3:5D4");
// GroupNames label

G:=SmallGroup(96,100);
// by ID

G=gap.SmallGroup(96,100);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,188,50,2309]);
// Polycyclic

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

Export

Character table of Dic35D4 in TeX

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