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G = D12.27D6  order 288 = 25·32

2nd non-split extension by D12 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D12.27D6, C12.83D12, C62.48D4, Dic6.27D6, C3⋊C8.27D6, C4○D12⋊1S3, C3⋊5(C4○D24), C6.68(C2×D12), (C2×C6).13D12, C32⋊9(C4○D8), C3⋊D24⋊15C2, (C2×C12).118D6, (C3×C12).117D4, C12.59D6⋊2C2, C3⋊1(Q8.13D6), C12.82(C3⋊D4), (C3×C12).65C23, (C6×C12).78C22, C32⋊3Q16⋊15C2, C32⋊5SD16⋊15C2, D12.S3⋊15C2, C4.32(C3⋊D12), C12.125(C22×S3), (C3×D12).35C22, C12⋊S3.26C22, C22.1(C3⋊D12), (C3×Dic6).34C22, C32⋊4Q8.26C22, (C2×C3⋊C8)⋊7S3, (C6×C3⋊C8)⋊10C2, C4.53(C2×S32), (C2×C4).66S32, C6.4(C2×C3⋊D4), (C3×C4○D12)⋊4C2, (C3×C6).69(C2×D4), C2.8(C2×C3⋊D12), (C3×C3⋊C8).30C22, (C2×C6).38(C3⋊D4), SmallGroup(288,477)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.27D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3 — D12.27D6
Lower central
C32 — C3×C6 — C3×C12 — D12.27D6
Upper central
C1 — C4 — C2×C4

Generators and relations for D12.27D6
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c5 >

Subgroups: 602 in 144 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C24⋊C2, D24, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×C24, C4○D12, C4○D12, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C4○D24, Q8.13D6, C3⋊D24, D12.S3, C32⋊5SD16, C32⋊3Q16, C6×C3⋊C8, C3×C4○D12, C12.59D6, D12.27D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C4○D8, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C4○D24, Q8.13D6, C2×C3⋊D12, D12.27D6

Smallest permutation representation of D12.27D6
►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 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

48 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J8A8B8C8D12A···12F12G···12K12L12M24A···24H
order12222333444446666666666888812···1212···12121224···24
size1121236224112123622224444121266662···24···412126···6

48 irreducible representations

dim1111111122222222222222444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D12C3⋊D4D12C3⋊D4C4○D8C4○D24S32C3⋊D12C2×S32C3⋊D12Q8.13D6D12.27D6
kernelD12.27D6C3⋊D24D12.S3C32⋊5SD16C32⋊3Q16C6×C3⋊C8C3×C4○D12C12.59D6C2×C3⋊C8C4○D12C3×C12C62C3⋊C8Dic6D12C2×C12C12C12C2×C6C2×C6C32C3C2×C4C4C4C22C3C1
# reps1111111111112112222248111124

Matrix representation of D12.27D6 ►in GL6(𝔽73)

010000
7200000
0072100
0072000
000010
000001
,
16160000
16570000
001000
0017200
000010
000001
,
2700000
0270000
0072000
0007200
000001
00007272
,
16570000
16160000
0072000
0007200
000001
000010

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,16,57,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

D12.27D6 in GAP, Magma, Sage, TeX 

D_{12}._{27}D_6
 
% in TeX
 
G:=Group("D12.27D6");
 
// GroupNames label
 
G:=SmallGroup(288,477);
 
// by ID
 
G=gap.SmallGroup(288,477);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,100,675,80,1356,9414]);
 
// 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,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^5>;
 
// generators/relations
 

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