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G = D12.15D4order 192 = 26·3

15th non-split extension by D12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.15D4, C42.86D6, Dic6.15D4, C4⋊Q88S3, C4.58(S3×D4), C12.41(C2×D4), (C2×C12).12D4, (C2×Q8).73D6, C6.54C22≀C2, C424S314C2, C12.10D46C2, Q8.11D63C2, C33(D4.10D4), (C6×Q8).67C22, C2.22(C232D6), (C4×C12).142C22, (C2×C12).413C23, C4○D12.22C22, Q8.15D6.2C2, C4.Dic3.15C22, (C3×C4⋊Q8)⋊8C2, (C2×C6).544(C2×D4), (C2×C4).11(C3⋊D4), C22.34(C2×C3⋊D4), (C2×C4).119(C22×S3), SmallGroup(192,654)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D12.15D4
C1C3C6C12C2×C12C4○D12Q8.15D6 — D12.15D4
C3C6C2×C12 — D12.15D4
C1C2C2×C4C4⋊Q8

Generators and relations for D12.15D4
 G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a10b, dcd=c3 >

Subgroups: 400 in 142 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×7], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×8], D4 [×6], Q8 [×8], Dic3 [×2], C12 [×2], C12 [×5], D6 [×2], C2×C6, C42, C4⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×6], C3⋊C8 [×2], Dic6 [×2], Dic6 [×2], C4×S3 [×6], D12 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C2×C12 [×2], C2×C12 [×2], C3×Q8 [×4], C4.10D4, C4≀C2 [×2], C4⋊Q8, C8.C22 [×2], 2- 1+4, C4.Dic3 [×2], Q82S3 [×2], C3⋊Q16 [×2], C4×C12, C3×C4⋊C4 [×2], C4○D12 [×2], C4○D12 [×2], S3×Q8 [×2], Q83S3 [×2], C6×Q8 [×2], D4.10D4, C424S3 [×2], C12.10D4, Q8.11D6 [×2], C3×C4⋊Q8, Q8.15D6, D12.15D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, D4.10D4, C232D6, D12.15D4

Character table of D12.15D4

 class 12A2B2C2D34A4B4C4D4E4F4G4H4I6A6B6C8A8B12A12B12C12D12E12F12G12H12I12J
 size 112121222244448121222224244444448888
ρ1111111111111111111111111111111    trivial
ρ21111-1111-111-1-1-11111-11111111-1-1-1-1    linear of order 2
ρ3111-11111-111-1-11-11111-1111111-1-1-1-1    linear of order 2
ρ4111-1-111111111-1-1111-1-11111111111    linear of order 2
ρ5111-11111-1-1-1-111-1111-11-1-1-111-11-11-1    linear of order 2
ρ6111-1-11111-1-11-1-1-111111-1-1-111-1-11-11    linear of order 2
ρ7111111111-1-11-111111-1-1-1-1-111-1-11-11    linear of order 2
ρ81111-1111-1-1-1-11-111111-1-1-1-111-11-11-1    linear of order 2
ρ922-20-222-20000020-22-200000-2200000    orthogonal lifted from D4
ρ1022-20222-200000-20-22-200000-2200000    orthogonal lifted from D4
ρ1122-2202-22000000-2-22-2000002-200000    orthogonal lifted from D4
ρ12222002-2-2200-200022200000-2-20020-2    orthogonal lifted from D4
ρ13222002-2-2-200200022200000-2-200-202    orthogonal lifted from D4
ρ1422-2-202-220000002-22-2000002-200000    orthogonal lifted from D4
ρ1522200-122-222-2-200-1-1-100-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1622200-1222222200-1-1-100-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1722200-122-2-2-2-2200-1-1-100111-1-11-11-11    orthogonal lifted from D6
ρ1822200-1222-2-22-200-1-1-100111-1-111-11-1    orthogonal lifted from D6
ρ1922200-1-2-2200-2000-1-1-100--3-3--311-3-3-1--31    complex lifted from C3⋊D4
ρ2022200-1-2-2200-2000-1-1-100-3--3-311--3--3-1-31    complex lifted from C3⋊D4
ρ2122200-1-2-2-2002000-1-1-100-3--3-311--3-31--3-1    complex lifted from C3⋊D4
ρ2222200-1-2-2-2002000-1-1-100--3-3--311-3--31-3-1    complex lifted from C3⋊D4
ρ2344-400-2-4400000002-2200000-2200000    orthogonal lifted from S3×D4
ρ2444-400-24-400000002-22000002-200000    orthogonal lifted from S3×D4
ρ254-40004000-2200000-4000-22200-20000    symplectic lifted from D4.10D4, Schur index 2
ρ264-400040002-200000-40002-2-20020000    symplectic lifted from D4.10D4, Schur index 2
ρ274-4000-20002-20000-2-322-300-1--31--31+-300-1+-30000    complex faithful
ρ284-4000-2000-220000-2-322-3001+-3-1+-3-1--3001--30000    complex faithful
ρ294-4000-2000-2200002-32-2-3001--3-1--3-1+-3001+-30000    complex faithful
ρ304-4000-20002-200002-32-2-300-1+-31+-31--300-1--30000    complex faithful

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

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

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

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

Matrix representation of D12.15D4 in GL4(𝔽7) generated by

3436
3531
5356
6511
,
2602
2360
3644
3635
,
2445
6336
3014
1251
,
3666
2304
2606
4241
G:=sub<GL(4,GF(7))| [3,3,5,6,4,5,3,5,3,3,5,1,6,1,6,1],[2,2,3,3,6,3,6,6,0,6,4,3,2,0,4,5],[2,6,3,1,4,3,0,2,4,3,1,5,5,6,4,1],[3,2,2,4,6,3,6,2,6,0,0,4,6,4,6,1] >;

D12.15D4 in GAP, Magma, Sage, TeX

D_{12}._{15}D_4
% in TeX

G:=Group("D12.15D4");
// GroupNames label

G:=SmallGroup(192,654);
// by ID

G=gap.SmallGroup(192,654);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,570,297,136,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^4=a^6,b*a*b=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^7*b,d*b*d=a^10*b,d*c*d=c^3>;
// generators/relations

Export

Character table of D12.15D4 in TeX

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