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G = D12.14D6  order 288 = 25·32

14th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.14D6, Dic6.12D6, C3⋊C8.19D6, Q8.16S32, C3⋊Q16⋊5S3, C6.65(S3×D4), Q8⋊2S3⋊5S3, (C3×Q8).34D6, D12⋊S3⋊6C2, C3⋊D24⋊10C2, C32⋊15(C4○D8), C3⋊Dic3.61D4, C3⋊3(D24⋊C2), C12.29D6⋊2C2, C3⋊4(Q8.7D6), C12.26D6⋊2C2, C12.27(C22×S3), (C3×C12).27C23, C32⋊5SD16⋊14C2, C2.25(Dic3⋊D6), (C3×D12).23C22, C12⋊S3.15C22, (Q8×C32).9C22, (C3×Dic6).22C22, C4.27(C2×S32), (C2×C3⋊S3).26D4, (C3×C3⋊Q16)⋊4C2, (C3×C3⋊C8).9C22, (C3×Q8⋊2S3)⋊8C2, (C3×C6).142(C2×D4), (C4×C3⋊S3).21C22, SmallGroup(288,598)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.14D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12.14D6
Lower central
C32 — C3×C6 — C3×C12 — D12.14D6
Upper central
C1 — C2 — C4 — Q8

Generators and relations for D12.14D6
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 666 in 144 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, S3×C8, C24⋊C2, D24, D4⋊S3, Q8⋊2S3, Q8⋊2S3, C3⋊Q16, C3×SD16, C3×Q16, D4⋊2S3, Q8⋊3S3, C3×C3⋊C8, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, Q8.7D6, D24⋊C2, C12.29D6, C3⋊D24, C32⋊5SD16, C3×Q8⋊2S3, C3×C3⋊Q16, D12⋊S3, C12.26D6, D12.14D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S32, S3×D4, C2×S32, Q8.7D6, D24⋊C2, Dic3⋊D6, D12.14D6

Smallest permutation representation of D12.14D6
►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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(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 14 3 16 5 18 7 20 9 22 11 24)(2 21 4 23 6 13 8 15 10 17 12 19)(25 42 35 40 33 38 31 48 29 46 27 44)(26 37 36 47 34 45 32 43 30 41 28 39)

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

33 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D8A8B8C8D12A12B12C···12G12H24A24B24C24D
order122223334444466668888121212···121224242424
size11121836224249912224246666448···82412121212

33 irreducible representations

dim111111112222222224444448
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C4○D8S32S3×D4C2×S32Q8.7D6D24⋊C2Dic3⋊D6D12.14D6
kernelD12.14D6C12.29D6C3⋊D24C32⋊5SD16C3×Q8⋊2S3C3×C3⋊Q16D12⋊S3C12.26D6Q8⋊2S3C3⋊Q16C3⋊Dic3C2×C3⋊S3C3⋊C8Dic6D12C3×Q8C32Q8C6C4C3C3C2C1
# reps111111111111211241212221

Matrix representation of D12.14D6 ►in GL6(𝔽73)

7230000
4810000
000100
00727200
000010
000001
,
0250000
3800000
0007200
0072000
000010
000001
,
2700000
18460000
0072000
0007200
0000072
000011
,
41480000
3800000
0072000
0007200
000001
000010

G:=sub<GL(6,GF(73))| [72,48,0,0,0,0,3,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,38,0,0,0,0,25,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,18,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,1],[41,38,0,0,0,0,48,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

D12.14D6 in GAP, Magma, Sage, TeX 

D_{12}._{14}D_6
 
% in TeX
 
G:=Group("D12.14D6");
 
// GroupNames label
 
G:=SmallGroup(288,598);
 
// by ID
 
G=gap.SmallGroup(288,598);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,100,675,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^6=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=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;
 
// generators/relations
 

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