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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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C4.Dic3, D4⋊S3, D4.S3, C4×C12, C3×C22⋊C4, C4○D12, C4○D12, S3×Q8, Q83S3, C6×D4, C6×Q8, D4.8D4, C424S3, C12.10D4, D126C22, C3×C4.4D4, Q8.15D6, D12.14D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, 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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 48)(9 47)(10 46)(11 45)(12 44)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)
(1 30 4 27 7 36 10 33)(2 29 5 26 8 35 11 32)(3 28 6 25 9 34 12 31)(13 47 16 44 19 41 22 38)(14 46 17 43 20 40 23 37)(15 45 18 42 21 39 24 48)
(1 27)(2 32)(3 25)(4 30)(5 35)(6 28)(7 33)(8 26)(9 31)(10 36)(11 29)(12 34)(13 44)(14 37)(15 42)(16 47)(17 40)(18 45)(19 38)(20 43)(21 48)(22 41)(23 46)(24 39)

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

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

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

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

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