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G = D12.32D6  order 288 = 25·32

7th non-split extension by D12 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D12.32D6, C62.46D4, Dic6.32D6, (C2×Dic6)⋊8S3, (C6×Dic6)⋊2C2, C4○D12.4S3, (C3×C12).65D4, (C2×C12).116D6, C32⋊2Q16⋊5C2, Dic6⋊S3⋊6C2, C3⋊4(Q8.14D6), C12.46(C3⋊D4), C12.82(C22×S3), (C3×C12).63C23, (C6×C12).76C22, C32⋊6(C8.C22), C12.58D6⋊10C2, C4.14(D6⋊S3), C3⋊4(Q8.11D6), (C3×D12).33C22, C32⋊4C8.5C22, C22.5(D6⋊S3), (C3×Dic6).33C22, (C2×C4).7S32, C4.76(C2×S32), (C3×C6).67(C2×D4), C6.73(C2×C3⋊D4), (C3×C4○D12).7C2, C2.8(C2×D6⋊S3), (C2×C6).18(C3⋊D4), SmallGroup(288,475)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.32D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — Dic6⋊S3 — D12.32D6
Lower central
C32 — C3×C6 — C3×C12 — D12.32D6
Upper central
C1 — C2 — C2×C4

Generators and relations for D12.32D6
 G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=a6c5 >

Subgroups: 402 in 130 conjugacy classes, 44 normal (34 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, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, Dic6, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×C12, S3×C6, C62, C4.Dic3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×Dic6, C4○D12, C6×Q8, C3×C4○D4, C32⋊4C8, C3×Dic6, C3×Dic6, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, Q8.11D6, Q8.14D6, Dic6⋊S3, C32⋊2Q16, C12.58D6, C6×Dic6, C3×C4○D12, D12.32D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, S32, C2×C3⋊D4, D6⋊S3, C2×S32, Q8.11D6, Q8.14D6, C2×D6⋊S3, D12.32D6

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

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J8A8B12A12B12C···12I12J···12O
order122233344444666666666688121212···1212···12
size11212224221212122222444412123636224···412···12

39 irreducible representations

dim11111122222222244444444
type+++++++++++++-+-+--
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C3⋊D4C3⋊D4C8.C22S32D6⋊S3C2×S32D6⋊S3Q8.11D6Q8.14D6D12.32D6
kernelD12.32D6Dic6⋊S3C32⋊2Q16C12.58D6C6×Dic6C3×C4○D12C2×Dic6C4○D12C3×C12C62Dic6D12C2×C12C12C2×C6C32C2×C4C4C4C22C3C3C1
# reps12211111113124411111224

Matrix representation of D12.32D6 ►in GL8(𝔽73)

172000000
10000000
007200000
000720000
00000100
000072000
000000072
00000010
,
5132000000
1022000000
0030600000
0013430000
000000072
00000010
00000100
000072000
,
10000000
01000000
00010000
0072720000
000007200
00001000
000000072
00000010
,
3013000000
6043000000
0070420000
004530000
0000113000
0000306200
0000003062
0000006243

G:=sub<GL(8,GF(73))| [1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[51,10,0,0,0,0,0,0,32,22,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[30,60,0,0,0,0,0,0,13,43,0,0,0,0,0,0,0,0,70,45,0,0,0,0,0,0,42,3,0,0,0,0,0,0,0,0,11,30,0,0,0,0,0,0,30,62,0,0,0,0,0,0,0,0,30,62,0,0,0,0,0,0,62,43] >;
 

D12.32D6 in GAP, Magma, Sage, TeX 

D_{12}._{32}D_6
 
% in TeX
 
G:=Group("D12.32D6");
 
// GroupNames label
 
G:=SmallGroup(288,475);
 
// by ID
 
G=gap.SmallGroup(288,475);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,100,675,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^5>;
 
// generators/relations
 

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