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G = D12.6D4order 192 = 26·3

6th non-split extension by D12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.6D4, Dic6.6D4, M4(2).5D6, C3⋊C8.26D4, D12.C47C2, C4.152(S3×D4), C12.99(C2×D4), C8⋊D6.1C2, (C2×Q8).31D6, C4.10D43S3, C32(D4.3D4), (C6×Q8).9C22, Q8.11D61C2, C12.53D43C2, C6.11(C4⋊D4), (C2×C12).11C23, C4○D12.7C22, C12.46D412C2, C2.14(Dic3⋊D4), (C2×D12).42C22, C4.Dic3.6C22, C22.15(C4○D12), (C3×M4(2)).22C22, (C2×C3⋊C8).3C22, (C2×Q82S3)⋊1C2, (C2×C6).32(C4○D4), (C3×C4.10D4)⋊1C2, (C2×C4).11(C22×S3), SmallGroup(192,313)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D12.6D4
C1C3C6C12C2×C12C4○D12D12.C4 — D12.6D4
C3C6C2×C12 — D12.6D4
C1C2C2×C4C4.10D4

Generators and relations for D12.6D4
 G = < a,b,c,d | a12=b2=1, c4=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a3c3 >

Subgroups: 336 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3 [×2], C6, C6, C8 [×5], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, Dic3, C12 [×2], C12, D6 [×3], C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8 [×2], C3⋊C8, C24 [×2], Dic6, C4×S3, D12, D12 [×2], C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2, D24, C2×C3⋊C8, C4.Dic3, Q82S3 [×3], C3⋊Q16, C3×M4(2) [×2], C2×D12, C4○D12, C6×Q8, D4.3D4, C12.53D4, C12.46D4, C3×C4.10D4, D12.C4, C8⋊D6, C2×Q82S3, Q8.11D6, D12.6D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4 [×2], D4.3D4, Dic3⋊D4, D12.6D4

Character table of D12.6D4

 class 12A2B2C2D34A4B4C4D6A6B8A8B8C8D8E8F8G12A12B12C12D24A24B24C24D
 size 11212242228122444668122444888888
ρ1111111111111111111111111111    trivial
ρ2111-1-1111-1-111-1-11111111-1-11-1-11    linear of order 2
ρ3111-1-11111-11111-1-11-1-111111111    linear of order 2
ρ411111111-1111-1-1-1-11-1-111-1-11-1-11    linear of order 2
ρ51111-11111111-1-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ6111-11111-1-11111-1-1-1-1111-1-1-111-1    linear of order 2
ρ7111-111111-111-1-111-11-11111-1-1-1-1    linear of order 2
ρ81111-1111-11111111-11-111-1-1-111-1    linear of order 2
ρ922200-12220-1-12200200-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-20022-2002-200220-20-22000000    orthogonal lifted from D4
ρ1122200-122-20-1-12200-200-1-1111-1-11    orthogonal lifted from D6
ρ1222-2-202-22022-200000002-2000000    orthogonal lifted from D4
ρ1322-20022-2002-200-2-2020-22000000    orthogonal lifted from D4
ρ1422200-12220-1-1-2-200-200-1-1-1-11111    orthogonal lifted from D6
ρ1522200-122-20-1-1-2-200200-1-111-111-1    orthogonal lifted from D6
ρ1622-2202-220-22-200000002-2000000    orthogonal lifted from D4
ρ17222002-2-20022-2i2i00000-2-20002i-2i0    complex lifted from C4○D4
ρ18222002-2-200222i-2i00000-2-2000-2i2i0    complex lifted from C4○D4
ρ1922200-1-2-200-1-12i-2i0000011-3--3-3i-i3    complex lifted from C4○D12
ρ2022200-1-2-200-1-12i-2i0000011--3-33i-i-3    complex lifted from C4○D12
ρ2122200-1-2-200-1-1-2i2i0000011-3--33-ii-3    complex lifted from C4○D12
ρ2222200-1-2-200-1-1-2i2i0000011--3-3-3-ii3    complex lifted from C4○D12
ρ2344-400-24-400-2200000002-2000000    orthogonal lifted from S3×D4
ρ2444-400-2-4400-220000000-22000000    orthogonal lifted from S3×D4
ρ254-400040000-40002-2-2-200000000000    complex lifted from D4.3D4
ρ264-400040000-4000-2-22-200000000000    complex lifted from D4.3D4
ρ278-8000-4000040000000000000000    orthogonal faithful

Smallest permutation representation of D12.6D4
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 28 10 31 7 34 4 25)(2 35 11 26 8 29 5 32)(3 30 12 33 9 36 6 27)(13 46 22 37 19 40 16 43)(14 41 23 44 20 47 17 38)(15 48 24 39 21 42 18 45)
(1 46 4 37 7 40 10 43)(2 47 5 38 8 41 11 44)(3 48 6 39 9 42 12 45)(13 34 16 25 19 28 22 31)(14 35 17 26 20 29 23 32)(15 36 18 27 21 30 24 33)

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

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

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

Matrix representation of D12.6D4 in GL6(𝔽73)

0720000
1720000
0017100
0017200
006712072
006010
,
7210000
010000
00006112
00721012
0000720
00670720
,
100000
010000
006112710
0000721
0035266
0036166
,
7200000
0720000
002716161
00172061
006712720
006112720

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,1,67,6,0,0,71,72,12,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,67,0,0,0,1,0,0,0,0,61,0,72,72,0,0,12,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,61,0,35,36,0,0,12,0,2,1,0,0,71,72,6,6,0,0,0,1,6,6],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,2,1,67,61,0,0,71,72,12,12,0,0,61,0,72,72,0,0,61,61,0,0] >;

D12.6D4 in GAP, Magma, Sage, TeX

D_{12}._6D_4
% in TeX

G:=Group("D12.6D4");
// GroupNames label

G:=SmallGroup(192,313);
// by ID

G=gap.SmallGroup(192,313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,184,297,136,1684,851,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^3>;
// generators/relations

Export

Character table of D12.6D4 in TeX

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