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G = D6⋊5D12  order 288 = 25·32

2nd semidirect product of D6 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D6⋊5D12, C62.93C23, D6⋊C4⋊7S3, (S3×C6)⋊4D4, (C2×C12)⋊1D6, C6.27(S3×D4), (C6×C12)⋊1C22, (C2×Dic3)⋊2D6, C6.29(C2×D12), C2.29(S3×D12), C32⋊3C22≀C2, C3⋊1(D6⋊D4), C6.D12⋊4C2, (C6×Dic3)⋊2C22, (C22×S3).46D6, C2.14(Dic3⋊D6), (C2×C4)⋊2S32, (C2×C3⋊S3)⋊3D4, (C3×D6⋊C4)⋊1C2, (C22×S32)⋊2C2, (C2×C12⋊S3)⋊2C2, (C2×C3⋊D12)⋊8C2, C22.128(C2×S32), (C3×C6).115(C2×D4), (S3×C2×C6).38C22, (C22×C3⋊S3)⋊1C22, (C2×C6).112(C22×S3), SmallGroup(288,571)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — D6⋊5D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C22×S32 — D6⋊5D12
Lower central
C32 — C62 — D6⋊5D12
Upper central
C1 — C22 — C2×C4

Generators and relations for D6⋊5D12
 G = < a,b,c,d | a6=b2=c12=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 1538 in 287 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22≀C2, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, D6⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, S3×C23, C3⋊D12, C6×Dic3, C12⋊S3, C6×C12, C2×S32, S3×C2×C6, C22×C3⋊S3, D6⋊D4, C6.D12, C3×D6⋊C4, C2×C3⋊D12, C2×C12⋊S3, C22×S32, D6⋊5D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, S32, C2×D12, S3×D4, C2×S32, D6⋊D4, S3×D12, Dic3⋊D6, D6⋊5D12

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222222223334446···6666666612···1212121212
size11116666181836224412122···2444121212124···412121212

42 irreducible representations

dim111111222222244444
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D12S32S3×D4C2×S32S3×D12Dic3⋊D6
kernelD6⋊5D12C6.D12C3×D6⋊C4C2×C3⋊D12C2×C12⋊S3C22×S32D6⋊C4S3×C6C2×C3⋊S3C2×Dic3C2×C12C22×S3D6C2×C4C6C22C2C2
# reps112211242222814142

Matrix representation of D6⋊5D12 ►in GL8(ℤ)

-10000000
0-1000000
00100000
00010000
00000100
0000-1-100
00000010
00000001
,
-1-2000000
01000000
00100000
00010000
00000-100
0000-1000
00000010
00000001
,
-1-2000000
11000000
000-10000
00100000
00001000
00000100
000000-11
000000-10
,
-10000000
11000000
00100000
000-10000
00001000
0000-1-100
000000-10
000000-11

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1] >;
 

D6⋊5D12 in GAP, Magma, Sage, TeX 

D_6\rtimes_5D_{12}
 
% in TeX
 
G:=Group("D6:5D12");
 
// GroupNames label
 
G:=SmallGroup(288,571);
 
// by ID
 
G=gap.SmallGroup(288,571);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,142,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^12=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

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