Copied to
clipboard

G = D12.14D4order 192 = 26·3

14th non-split extension by D12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.14D4, C42.66D6, Dic6.14D4, C4.52(S3×D4), (C2×D4).52D6, C12.29(C2×D4), C4.4D45S3, (C2×C12).11D4, (C2×Q8).65D6, C6.52C22≀C2, D126C222C2, C33(D4.8D4), C424S312C2, C12.10D45C2, Q8.15D62C2, (C6×D4).68C22, (C6×Q8).59C22, C2.20(C232D6), (C4×C12).110C22, (C2×C12).380C23, C4○D12.20C22, C4.Dic3.14C22, (C2×C6).511(C2×D4), (C3×C4.4D4)⋊5C2, (C2×C4).10(C3⋊D4), C22.32(C2×C3⋊D4), (C2×C4).117(C22×S3), SmallGroup(192,621)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D12.14D4
C1C3C6C12C2×C12C4○D12Q8.15D6 — D12.14D4
C3C6C2×C12 — D12.14D4
C1C2C2×C4C4.4D4

Generators and relations for D12.14D4
 G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a10b, dcd=a6c3 >

Subgroups: 432 in 146 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], D4 [×8], Q8 [×6], C23, Dic3 [×2], C12 [×2], C12 [×3], D6 [×2], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×Q8, 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, C3×D4 [×2], C3×Q8 [×2], C22×C6, C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22 [×2], 2- 1+4, C4.Dic3 [×2], D4⋊S3 [×2], D4.S3 [×2], C4×C12, C3×C22⋊C4 [×2], C4○D12 [×2], C4○D12 [×2], S3×Q8 [×2], Q83S3 [×2], C6×D4, C6×Q8, D4.8D4, C424S3 [×2], C12.10D4, D126C22 [×2], C3×C4.4D4, Q8.15D6, D12.14D4
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, D4.8D4, C232D6, D12.14D4

Character table of D12.14D4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C6D6E8A8B12A12B12C12D12E12F12G12H
 size 112812122224444121222288242444444488
ρ1111111111111111111111111111111    trivial
ρ2111-11-11111-11-11-1111-1-1-11111111-1-1    linear of order 2
ρ3111-111111-11-1111111-1-1-1-1-1-111-1-111    linear of order 2
ρ411111-1111-1-1-1-11-1111111-1-1-111-1-1-1-1    linear of order 2
ρ5111-1-111111-11-1-11111-1-11-1111111-1-1    linear of order 2
ρ61111-1-11111111-1-111111-1-111111111    linear of order 2
ρ71111-11111-1-1-1-1-1111111-11-1-111-1-1-1-1    linear of order 2
ρ8111-1-1-1111-11-11-1-1111-1-111-1-111-1-111    linear of order 2
ρ9222-200-1222-22-200-1-1-11100-1-1-1-1-1-111    orthogonal lifted from D6
ρ1022-20-202-22000020-22-2000000-220000    orthogonal lifted from D4
ρ1122-20202-220000-20-22-2000000-220000    orthogonal lifted from D4
ρ1222-200-222-2000002-22-20000002-20000    orthogonal lifted from D4
ρ13222200-122-2-2-2-200-1-1-1-1-10011-1-11111    orthogonal lifted from D6
ρ1422-200222-200000-2-22-20000002-20000    orthogonal lifted from D4
ρ152220002-2-2020-200222000000-2-2002-2    orthogonal lifted from D4
ρ162220002-2-20-20200222000000-2-200-22    orthogonal lifted from D4
ρ17222200-122222200-1-1-1-1-100-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ18222-200-122-22-2200-1-1-1110011-1-111-1-1    orthogonal lifted from D6
ρ19222000-1-2-2020-200-1-1-1-3--300--3-311--3-3-11    complex lifted from C3⋊D4
ρ20222000-1-2-20-20200-1-1-1-3--300-3--311-3--31-1    complex lifted from C3⋊D4
ρ21222000-1-2-2020-200-1-1-1--3-300-3--311-3--3-11    complex lifted from C3⋊D4
ρ22222000-1-2-20-20200-1-1-1--3-300--3-311--3-31-1    complex lifted from C3⋊D4
ρ2344-4000-24-40000002-22000000-220000    orthogonal lifted from S3×D4
ρ2444-4000-2-440000002-220000002-20000    orthogonal lifted from S3×D4
ρ254-40000400-2i02i0000-400000-2i-2i002i2i00    complex lifted from D4.8D4
ρ264-400004002i0-2i0000-4000002i2i00-2i-2i00    complex lifted from D4.8D4
ρ274-40000-2002i0-2i0002-32-2-300004ζ324ζ30043ζ3243ζ300    complex faithful
ρ284-40000-200-2i02i000-2-322-3000043ζ343ζ32004ζ34ζ3200    complex faithful
ρ294-40000-200-2i02i0002-32-2-3000043ζ3243ζ3004ζ324ζ300    complex faithful
ρ304-40000-2002i0-2i000-2-322-300004ζ34ζ320043ζ343ζ3200    complex faithful

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

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

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

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

Matrix representation of D12.14D4 in GL4(𝔽73) generated by

0900
64000
00065
0080
,
00700
0003
24000
04900
,
00072
00720
1000
07200
,
0001
00720
07200
1000
G:=sub<GL(4,GF(73))| [0,64,0,0,9,0,0,0,0,0,0,8,0,0,65,0],[0,0,24,0,0,0,0,49,70,0,0,0,0,3,0,0],[0,0,1,0,0,0,0,72,0,72,0,0,72,0,0,0],[0,0,0,1,0,0,72,0,0,72,0,0,1,0,0,0] >;

D12.14D4 in GAP, Magma, Sage, TeX

D_{12}._{14}D_4
% in TeX

G:=Group("D12.14D4");
// GroupNames label

G:=SmallGroup(192,621);
// by ID

G=gap.SmallGroup(192,621);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,1123,570,297,136,1684,6278]);
// 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=a^6*c^3>;
// generators/relations

Export

Character table of D12.14D4 in TeX

׿
×
𝔽