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G = S3×C4○D12  order 288 = 25·32

Direct product of S3 and C4○D12

direct product, metabelian, supersoluble, monomial

Aliases: S3×C4○D12, D12⋊26D6, Dic6⋊25D6, C62.137C23, (C4×S3)⋊15D6, (C2×C12)⋊22D6, C3⋊D4⋊11D6, (S3×D12)⋊13C2, (C6×C12)⋊5C22, (C2×Dic3)⋊20D6, (S3×Dic6)⋊13C2, D6.3D6⋊7C2, D6.D6⋊3C2, (S3×C6).4C23, (C3×C6).12C24, C6.12(S3×C23), D12⋊5S3⋊13C2, D6.5(C22×S3), (S3×C12)⋊15C22, D6.6D6⋊13C2, C12.59D6⋊8C2, (C3×D12)⋊29C22, (C22×S3).73D6, C6.D6⋊6C22, C32⋊7D4⋊6C22, C12⋊S3⋊22C22, D6⋊S3⋊12C22, C3⋊D12⋊13C22, (C3×C12).117C23, C12.150(C22×S3), (S3×Dic3)⋊12C22, (C3×Dic6)⋊28C22, (C6×Dic3)⋊28C22, C32⋊2Q8⋊11C22, C3⋊Dic3.17C23, (C3×Dic3).7C23, C32⋊4Q8⋊21C22, Dic3.20(C22×S3), (C2×C4)⋊7S32, (C4×S32)⋊2C2, (S3×C2×C4)⋊6S3, C4.97(C2×S32), C3⋊1(S3×C4○D4), (S3×C2×C12)⋊10C2, C3⋊4(C2×C4○D12), (S3×C3⋊D4)⋊7C2, C22.7(C2×S32), C32⋊5(C2×C4○D4), (C2×S32).9C22, C2.14(C22×S32), (C3×C4○D12)⋊11C2, (C3×S3)⋊1(C4○D4), (C4×C3⋊S3)⋊10C22, (C3×C3⋊D4)⋊6C22, (C2×C3⋊S3).19C23, (S3×C2×C6).107C22, (C2×C6).154(C22×S3), SmallGroup(288,953)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — S3×C4○D12
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — S3×C4○D12
Lower central
C32 — C3×C6 — S3×C4○D12
Upper central
C1 — C4 — C2×C4

Generators and relations for S3×C4○D12
 G = < a,b,c,d,e | a3=b2=c4=e2=1, d6=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >

Subgroups: 1242 in 348 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C4○D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, S3×C6, C2×C3⋊S3, C62, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C22×C12, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C2×S32, S3×C2×C6, C2×C4○D12, S3×C4○D4, S3×Dic6, D12⋊5S3, D6.D6, D6.6D6, C4×S32, S3×D12, D6.3D6, S3×C3⋊D4, S3×C2×C12, C3×C4○D12, C12.59D6, S3×C4○D12
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, C4○D12, S3×C23, C2×S32, C2×C4○D12, S3×C4○D4, C22×S32, S3×C4○D12

Smallest permutation representation of S3×C4○D12
►On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E6F6G6H6I6J6K6L6M6N12A···12F12G···12K12L12M12N12O12P12Q
order122222222233344444444446666666666666612···1212···12121212121212
size11233666181822411233666181822224444666612122···24···466661212

54 irreducible representations

dim1111111111112222222222244444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3S3D6D6D6D6D6D6D6C4○D4C4○D12S32C2×S32C2×S32S3×C4○D4S3×C4○D12
kernelS3×C4○D12S3×Dic6D12⋊5S3D6.D6D6.6D6C4×S32S3×D12D6.3D6S3×C3⋊D4S3×C2×C12C3×C4○D12C12.59D6S3×C2×C4C4○D12Dic6C4×S3D12C2×Dic3C3⋊D4C2×C12C22×S3C3×S3S3C2×C4C4C22C3C1
# reps1112121221111116112214812124

Matrix representation of S3×C4○D12 ►in GL6(𝔽13)

010000
12120000
001000
000100
000010
000001
,
100000
12120000
0012000
0001200
0000120
0000012
,
1200000
0120000
005000
000500
0000120
0000012
,
1200000
0120000
000800
008000
000011
0000120
,
1200000
0120000
000800
005000
00001212
000001

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;
 

S3×C4○D12 in GAP, Magma, Sage, TeX 

S_3\times C_4\circ D_{12}
 
% in TeX
 
G:=Group("S3xC4oD12");
 
// GroupNames label
 
G:=SmallGroup(288,953);
 
// by ID
 
G=gap.SmallGroup(288,953);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,346,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^2=1,d^6=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^5>;
 
// generators/relations
 

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