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G = D63Q8order 96 = 25·3

3rd semidirect product of D6 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D63Q8, C12.22D4, (C6×Q8)⋊3C2, (C2×Q8)⋊5S3, C2.9(S3×Q8), D6⋊C4.6C2, C6.57(C2×D4), (C2×C4).21D6, C35(C22⋊Q8), C6.17(C2×Q8), C4⋊Dic315C2, Dic3⋊C416C2, C6.36(C4○D4), C4.18(C3⋊D4), (C2×C6).58C23, (C2×C12).64C22, C2.8(Q83S3), C22.64(C22×S3), (C22×S3).27C22, (C2×Dic3).21C22, (S3×C2×C4).5C2, C2.21(C2×C3⋊D4), SmallGroup(96,153)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D63Q8
C1C3C6C2×C6C22×S3S3×C2×C4 — D63Q8
C3C2×C6 — D63Q8
C1C22C2×Q8

Generators and relations for D63Q8
 G = < a,b,c,d | a6=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 162 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22⋊Q8, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, D63Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, S3×Q8, Q83S3, C2×C3⋊D4, D63Q8

Character table of D63Q8

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C12A12B12C12D12E12F
 size 11116622244661212222444444
ρ1111111111111111111111111    trivial
ρ2111111111-1-111-1-11111-1-11-1-1    linear of order 2
ρ31111-1-11-1-11-1111-1111-11-1-1-11    linear of order 2
ρ41111-1-11-1-1-1111-11111-1-11-11-1    linear of order 2
ρ51111-1-1111-1-1-1-1111111-1-11-1-1    linear of order 2
ρ61111-1-111111-1-1-1-1111111111    linear of order 2
ρ71111111-1-1-11-1-11-1111-1-11-11-1    linear of order 2
ρ81111111-1-11-1-1-1-11111-11-1-1-11    linear of order 2
ρ9222200-1-2-2-220000-1-1-111-11-11    orthogonal lifted from D6
ρ1022-2-20022-2000000-22-2-200200    orthogonal lifted from D4
ρ11222200-122-2-20000-1-1-1-111-111    orthogonal lifted from D6
ρ1222-2-2002-22000000-22-2200-200    orthogonal lifted from D4
ρ13222200-122220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ14222200-1-2-22-20000-1-1-11-1111-1    orthogonal lifted from D6
ρ152-22-22-22000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ162-22-2-222000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ1722-2-200-1-220000001-11-1--3--31-3-3    complex lifted from C3⋊D4
ρ1822-2-200-12-20000001-111--3-3-1--3-3    complex lifted from C3⋊D4
ρ1922-2-200-1-220000001-11-1-3-31--3--3    complex lifted from C3⋊D4
ρ2022-2-200-12-20000001-111-3--3-1-3--3    complex lifted from C3⋊D4
ρ212-2-2200200002i-2i00-2-22000000    complex lifted from C4○D4
ρ222-2-220020000-2i2i00-2-22000000    complex lifted from C4○D4
ρ234-4-4400-20000000022-2000000    orthogonal lifted from Q83S3, Schur index 2
ρ244-44-400-200000000-222000000    symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of D63Q8
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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 44)(8 43)(9 48)(10 47)(11 46)(12 45)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)
(1 29 17 22)(2 30 18 23)(3 25 13 24)(4 26 14 19)(5 27 15 20)(6 28 16 21)(7 32 44 39)(8 33 45 40)(9 34 46 41)(10 35 47 42)(11 36 48 37)(12 31 43 38)
(1 41 17 34)(2 42 18 35)(3 37 13 36)(4 38 14 31)(5 39 15 32)(6 40 16 33)(7 27 44 20)(8 28 45 21)(9 29 46 22)(10 30 47 23)(11 25 48 24)(12 26 43 19)

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

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

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

D63Q8 is a maximal subgroup of
D6.1SD16  D62SD16  D6.Q16  C3⋊(C8⋊D4)  D61Q16  D6⋊C8.C2  C8⋊Dic3⋊C2  C3⋊C8.D4  D66SD16  C2414D4  Dic6.16D4  C248D4  D65Q16  D12.17D4  D63Q16  C24.36D4  C42.232D6  D1210Q8  C42.131D6  C42.132D6  C42.133D6  C42.134D6  C42.135D6  S3×C22⋊Q8  C4⋊C426D6  C6.162- 1+4  C6.172- 1+4  C6.512+ 1+4  C6.1182+ 1+4  C6.522+ 1+4  C6.532+ 1+4  C6.202- 1+4  C6.212- 1+4  C6.232- 1+4  C6.772- 1+4  C6.782- 1+4  C6.252- 1+4  C6.592+ 1+4  C42.137D6  D1210D4  Dic610D4  C4222D6  C4223D6  C42.234D6  C42.144D6  C42.145D6  D1212D4  D128Q8  C42.241D6  C42.174D6  D129Q8  C42.176D6  C42.178D6  C42.180D6  Q8×C3⋊D4  C6.442- 1+4  C6.452- 1+4  C6.1042- 1+4  (C2×D4)⋊43D6  C6.1452+ 1+4  C6.1082- 1+4  D183Q8  D66Dic6  C12.30D12  D61Dic6  C62.58C23  C62.261C23  C60.46D4  D3010Q8  D61Dic10  D30⋊Q8  D307Q8
D63Q8 is a maximal quotient of
C12⋊(C4⋊C4)  C6.67(C4×D4)  (C2×C12).54D4  (C2×C12).288D4  C4⋊(D6⋊C4)  D6⋊C46C4  (C2×C12).290D4  (C2×C12).56D4  Dic6.4Q8  D12.4Q8  D125Q8  D126Q8  Dic65Q8  Dic66Q8  (C6×Q8)⋊7C4  C22.52(S3×Q8)  (C22×Q8)⋊9S3  D183Q8  D66Dic6  C12.30D12  D61Dic6  C62.58C23  C62.261C23  C60.46D4  D3010Q8  D61Dic10  D30⋊Q8  D307Q8

Matrix representation of D63Q8 in GL6(𝔽13)

1200000
0120000
0001200
0011200
0000120
0000012
,
1200000
910000
0011200
0001200
000010
0000712
,
730000
1060000
001000
000100
000093
000034
,
1200000
0120000
0012000
0001200
000050
000098

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,7,0,0,0,0,0,12],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,3,4],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,9,0,0,0,0,0,8] >;

D63Q8 in GAP, Magma, Sage, TeX

D_6\rtimes_3Q_8
% in TeX

G:=Group("D6:3Q8");
// GroupNames label

G:=SmallGroup(96,153);
// by ID

G=gap.SmallGroup(96,153);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,86,2309]);
// Polycyclic

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

Export

Character table of D63Q8 in TeX

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