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G = D6⋊4D12  order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: D6⋊4D12, C62.92C23, (S3×C6)⋊3D4, D6⋊C4⋊18S3, (C2×D12)⋊6S3, (C2×C12)⋊13D6, C6.26(S3×D4), (C6×D12)⋊23C2, D6⋊5(C3⋊D4), (C2×Dic3)⋊1D6, C6.28(C2×D12), C2.28(S3×D12), C32⋊2C22≀C2, (C22×S3)⋊2D6, C3⋊3(D6⋊D4), C3⋊1(C23⋊2D6), D6⋊Dic3⋊14C2, (C6×C12)⋊17C22, (C6×Dic3)⋊1C22, C6.11D12⋊12C2, C2.20(D6⋊D6), (C2×C4)⋊1S32, (C2×C3⋊S3)⋊2D4, (C22×S32)⋊1C2, (S3×C2×C6)⋊1C22, (C3×D6⋊C4)⋊15C2, C2.19(S3×C3⋊D4), C6.41(C2×C3⋊D4), (C2×D6⋊S3)⋊6C2, (C2×C3⋊D12)⋊7C2, C22.127(C2×S32), (C3×C6).114(C2×D4), (C2×C3⋊Dic3)⋊2C22, (C2×C6).111(C22×S3), (C22×C3⋊S3).27C22, SmallGroup(288,570)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — D6⋊4D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C22×S32 — D6⋊4D12
Lower central
C32 — C62 — D6⋊4D12
Upper central
C1 — C22 — C2×C4

Generators and relations for D6⋊4D12
 G = < a,b,c,d | a6=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c-1 >

Subgroups: 1370 in 277 conjugacy classes, 54 normal (44 characteristic)
C1, 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, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22≀C2, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×D12, C2×D12, C2×C3⋊D4, C6×D4, S3×C23, D6⋊S3, C3⋊D12, C3×D12, C6×Dic3, C2×C3⋊Dic3, C6×C12, C2×S32, S3×C2×C6, C22×C3⋊S3, D6⋊D4, C23⋊2D6, D6⋊Dic3, C3×D6⋊C4, C6.11D12, C2×D6⋊S3, C2×C3⋊D12, C6×D12, C22×S32, D6⋊4D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C22≀C2, S32, C2×D12, S3×D4, C2×C3⋊D4, C2×S32, D6⋊D4, C23⋊2D6, S3×D12, D6⋊D6, S3×C3⋊D4, D6⋊4D12

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6F6G6H6I6J···6O12A···12H12I12J
order122222222223334446···66666···612···121212
size11116666121818224412362···244412···124···41212

42 irreducible representations

dim11111111222222222444444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D12C3⋊D4S32S3×D4C2×S32S3×D12D6⋊D6S3×C3⋊D4
kernelD6⋊4D12D6⋊Dic3C3×D6⋊C4C6.11D12C2×D6⋊S3C2×C3⋊D12C6×D12C22×S32D6⋊C4C2×D12S3×C6C2×C3⋊S3C2×Dic3C2×C12C22×S3D6D6C2×C4C6C22C2C2C2
# reps11111111114212344141222

Matrix representation of D6⋊4D12 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
000000120
000000012
,
012000000
120000000
00100000
00010000
000012000
000012100
000000120
000000121
,
80000000
05000000
001120000
00100000
000012000
000001200
000000111
000000012
,
05000000
80000000
00100000
001120000
000012000
000001200
000000111
000000012

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,12],[0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,12] >;
 

D6⋊4D12 in GAP, Magma, Sage, TeX 

D_6\rtimes_4D_{12}
 
% in TeX
 
G:=Group("D6:4D12");
 
// GroupNames label
 
G:=SmallGroup(288,570);
 
// by ID
 
G=gap.SmallGroup(288,570);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^-1>;
 
// generators/relations
 

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