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G = Q85D12order 192 = 26·3

3rd semidirect product of Q8 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D44D12, Q85D12, C424D6, D1215D4, Dic615D4, M4(2)⋊3D6, C4≀C21S3, (C3×D4)⋊3D4, (C3×Q8)⋊3D4, D4○D121C2, C8⋊D68C2, C4.9(C2×D12), D4⋊D61C2, C4⋊D126C2, C32(D44D4), C4○D4.17D6, C4.125(S3×D4), C424S35C2, (C4×C12)⋊11C22, C6.27C22≀C2, C12.337(C2×D4), (C22×S3).2D4, C22.29(S3×D4), C12.46D41C2, (C2×D12)⋊13C22, C4.Dic34C22, C2.30(D6⋊D4), (C2×C12).262C23, C4○D12.11C22, (C3×M4(2))⋊10C22, (C3×C4≀C2)⋊1C2, (C2×C6).26(C2×D4), (C3×C4○D4).3C22, (C2×C4).109(C22×S3), SmallGroup(192,381)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Q85D12
C1C3C6C2×C6C2×C12C2×D12D4○D12 — Q85D12
C3C6C2×C12 — Q85D12
C1C2C2×C4C4≀C2

Generators and relations for Q85D12
 G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 672 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×11], S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4, D4 [×15], Q8, Q8, C23 [×5], Dic3, C12 [×2], C12 [×3], D6 [×10], C2×C6, C2×C6, C42, M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4 [×8], C4○D4, C4○D4 [×3], C3⋊C8, C24, Dic6, C4×S3 [×3], D12, D12 [×10], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3 [×2], C22×S3 [×3], C4.D4, C4≀C2, C4≀C2, C41D4, C8⋊C22 [×2], 2+ 1+4, C24⋊C2, D24, C4.Dic3, D4⋊S3, Q82S3, C4×C12, C3×M4(2), C2×D12 [×2], C2×D12 [×3], C4○D12, C4○D12, S3×D4 [×3], Q83S3, C3×C4○D4, D44D4, C424S3, C12.46D4, C3×C4≀C2, C4⋊D12, C8⋊D6, D4⋊D6, D4○D12, Q85D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], D44D4, D6⋊D4, Q85D12

Character table of Q85D12

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C8A8B12A12B12C12D12E12F12G12H24A24B
 size 112412121224222444122488242244444888
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-11111-1-11-1111-1111-1-1-1-111-1-1    linear of order 2
ρ3111-1-111111111-1-111-1-1-11111111-1-1-1    linear of order 2
ρ4111-11-1-11111-1-1-1111-11-111-1-1-1-11-111    linear of order 2
ρ5111-1-111-1111-1-1-1-111-11111-1-1-1-11-111    linear of order 2
ρ6111-11-1-1-111111-1111-1-111111111-1-1-1    linear of order 2
ρ71111111-1111-1-111111-1-111-1-1-1-111-1-1    linear of order 2
ρ81111-1-1-1-1111111-11111-11111111111    linear of order 2
ρ922220000-122-2-220-1-1-1-20-1-11111-1-111    orthogonal lifted from D6
ρ10222002-202-2-2000022000-2-20000-2000    orthogonal lifted from D4
ρ1122-2-2000022-200202-2-200-2-200002200    orthogonal lifted from D4
ρ1222-2020002-22000-22-2000220000-2000    orthogonal lifted from D4
ρ1322-22000022-200-202-2200-2-200002-200    orthogonal lifted from D4
ρ1422-20-20002-2200022-2000220000-2000    orthogonal lifted from D4
ρ15222-20000-12222-20-1-11-20-1-1-1-1-1-1-1111    orthogonal lifted from D6
ρ1622200-2202-2-2000022000-2-20000-2000    orthogonal lifted from D4
ρ17222-20000-122-2-2-20-1-1120-1-11111-11-1-1    orthogonal lifted from D6
ρ1822220000-1222220-1-1-120-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1922-220000-12-200-20-11-1001133-3-3-113-3    orthogonal lifted from D12
ρ2022-2-20000-12-20020-111001133-3-3-1-1-33    orthogonal lifted from D12
ρ2122-220000-12-200-20-11-10011-3-333-11-33    orthogonal lifted from D12
ρ2222-2-20000-12-20020-1110011-3-333-1-13-3    orthogonal lifted from D12
ρ234-4000000400-2200-40000002-2-220000    orthogonal lifted from D44D4
ρ2444-400000-2-440000-22000-2-200002000    orthogonal lifted from S3×D4
ρ2544400000-2-4-40000-2-20002200002000    orthogonal lifted from S3×D4
ρ264-40000004002-200-4000000-222-20000    orthogonal lifted from D44D4
ρ274-4000000-2002-2002000023-231+3-1-3-1+31-30000    orthogonal faithful
ρ284-4000000-200-220020000-2323-1+31-31+3-1-30000    orthogonal faithful
ρ294-4000000-200-22002000023-23-1-31+31-3-1+30000    orthogonal faithful
ρ304-4000000-2002-20020000-23231-3-1+3-1-31+30000    orthogonal faithful

Permutation representations of Q85D12
On 24 points - transitive group 24T365
Generators in S24
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 19 16)(14 23 20 17)(15 24 21 18)
(1 22 4 16)(2 20 5 14)(3 18 6 24)(7 19 10 13)(8 17 11 23)(9 15 12 21)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 6)(2 5)(3 4)(7 9)(10 12)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)

G:=sub<Sym(24)| (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,19,16)(14,23,20,17)(15,24,21,18), (1,22,4,16)(2,20,5,14)(3,18,6,24)(7,19,10,13)(8,17,11,23)(9,15,12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,9)(10,12)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;

G:=Group( (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,19,16)(14,23,20,17)(15,24,21,18), (1,22,4,16)(2,20,5,14)(3,18,6,24)(7,19,10,13)(8,17,11,23)(9,15,12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,9)(10,12)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22) );

G=PermutationGroup([(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,19,16),(14,23,20,17),(15,24,21,18)], [(1,22,4,16),(2,20,5,14),(3,18,6,24),(7,19,10,13),(8,17,11,23),(9,15,12,21)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6),(2,5),(3,4),(7,9),(10,12),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22)])

G:=TransitiveGroup(24,365);

Matrix representation of Q85D12 in GL4(𝔽73) generated by

71400
596600
006659
00147
,
0010
0001
72000
07200
,
727200
1000
00766
00714
,
727200
0100
00766
005966
G:=sub<GL(4,GF(73))| [7,59,0,0,14,66,0,0,0,0,66,14,0,0,59,7],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[72,1,0,0,72,0,0,0,0,0,7,7,0,0,66,14],[72,0,0,0,72,1,0,0,0,0,7,59,0,0,66,66] >;

Q85D12 in GAP, Magma, Sage, TeX

Q_8\rtimes_5D_{12}
% in TeX

G:=Group("Q8:5D12");
// GroupNames label

G:=SmallGroup(192,381);
// by ID

G=gap.SmallGroup(192,381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,570,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of Q85D12 in TeX

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