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G = D6⋊D12  order 288 = 25·32

1st semidirect product of D6 and D12 acting via D12/C12=C2

metabelian, supersoluble, monomial

Aliases: D6⋊1D12, C62.76C23, D6⋊C4⋊3S3, (S3×C6)⋊8D4, C6.20(S3×D4), C6.21(C2×D12), C2.22(S3×D12), C3⋊4(Dic3⋊D4), C3⋊2(C12⋊7D4), (C3×Dic3)⋊11D4, (C2×C12).262D6, C32⋊5(C4⋊D4), Dic3⋊Dic3⋊8C2, C6.33(C4○D12), Dic3⋊4(C3⋊D4), (C2×Dic3).30D6, (C22×S3).12D6, C6.11D12⋊17C2, (C6×C12).237C22, C2.17(D6.D6), (C6×Dic3).15C22, (S3×C2×C12)⋊1C2, (S3×C2×C4)⋊13S3, (C2×C4).51S32, (C3×D6⋊C4)⋊6C2, C6.38(C2×C3⋊D4), C2.17(S3×C3⋊D4), (C2×D6⋊S3)⋊2C2, (C2×C3⋊D12)⋊2C2, C22.114(C2×S32), (C3×C6).103(C2×D4), (S3×C2×C6).27C22, (C3×C6).46(C4○D4), (C2×C6).95(C22×S3), (C22×C3⋊S3).21C22, (C2×C3⋊Dic3).53C22, SmallGroup(288,554)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — D6⋊D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — D6⋊D12
Lower central
C32 — C62 — D6⋊D12
Upper central
C1 — C22 — C2×C4

Generators and relations for D6⋊D12
 G = < a,b,c,d | a6=b2=c12=d2=1, bab=cac-1=dad=a-1, cbc-1=a4b, dbd=ab, dcd=c-1 >

Subgroups: 922 in 205 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C4⋊D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, D6⋊S3, C3⋊D12, S3×C12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, Dic3⋊D4, C12⋊7D4, Dic3⋊Dic3, C3×D6⋊C4, C6.11D12, C2×D6⋊S3, C2×C3⋊D12, S3×C2×C12, D6⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, S32, C2×D12, C4○D12, S3×D4, C2×C3⋊D4, C2×S32, Dic3⋊D4, C12⋊7D4, D6.D6, S3×D12, S3×C3⋊D4, D6⋊D12

Smallest permutation representation of D6⋊D12
►On 48 points
Generators in S48
(1 48 5 40 9 44)(2 45 10 41 6 37)(3 38 7 42 11 46)(4 47 12 43 8 39)(13 30 17 34 21 26)(14 27 22 35 18 31)(15 32 19 36 23 28)(16 29 24 25 20 33)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M6N6O12A12B12C12D12E···12J12K12L12M12N12O12P
order122222223334444446···66666666661212121212···12121212121212
size1111661236224226612362···24446666121222224···466661212

48 irreducible representations

dim111111122222222222444444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4C3⋊D4D12C4○D12S32S3×D4C2×S32D6.D6S3×D12S3×C3⋊D4
kernelD6⋊D12Dic3⋊Dic3C3×D6⋊C4C6.11D12C2×D6⋊S3C2×C3⋊D12S3×C2×C12D6⋊C4S3×C2×C4C3×Dic3S3×C6C2×Dic3C2×C12C22×S3C3×C6Dic3D6C6C2×C4C6C22C2C2C2
# reps111112111222222448121222

Matrix representation of D6⋊D12 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
96000000
44000000
001200000
000120000
00001000
000011200
00000010
00000001
,
10000000
01000000
00010000
001200000
000001200
000012000
00000001
0000001212
,
120000000
31000000
00100000
000120000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,4,0,0,0,0,0,0,6,4,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

D6⋊D12 in GAP, Magma, Sage, TeX 

D_6\rtimes D_{12}
 
% in TeX
 
G:=Group("D6:D12");
 
// GroupNames label
 
G:=SmallGroup(288,554);
 
// by ID
 
G=gap.SmallGroup(288,554);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,142,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^4*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

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