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G = C2×D12⋊S3  order 288 = 25·32

Direct product of C2 and D12⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C2×D12⋊S3, D12⋊21D6, Dic6⋊20D6, C62.128C23, (C2×D12)⋊12S3, (C6×D12)⋊14C2, (C3×C6).3C24, C6.3(S3×C23), C6⋊1(D4⋊2S3), C6⋊2(Q8⋊3S3), (C6×Dic6)⋊18C2, (C2×Dic6)⋊12S3, (C2×C12).165D6, (S3×C6).1C23, D6.2(C22×S3), (C3×D12)⋊26C22, (C2×Dic3).87D6, (S3×Dic3)⋊5C22, (C22×S3).54D6, C3⋊D12⋊11C22, (C6×C12).158C22, (C3×C12).112C23, C12.107(C22×S3), (C3×Dic6)⋊25C22, C3⋊Dic3.36C23, (C3×Dic3).2C23, Dic3.1(C22×S3), (C6×Dic3).44C22, C4.78(C2×S32), (C2×C4).122S32, C2.6(C22×S32), (C2×S3×Dic3)⋊3C2, C32⋊2(C2×C4○D4), (C3×C6)⋊2(C4○D4), C3⋊1(C2×D4⋊2S3), C22.60(C2×S32), C3⋊2(C2×Q8⋊3S3), (C4×C3⋊S3)⋊11C22, (S3×C2×C6).64C22, (C2×C3⋊D12)⋊18C2, (C2×C3⋊S3).40C23, (C2×C6).147(C22×S3), (C22×C3⋊S3).99C22, (C2×C3⋊Dic3).177C22, (C2×C4×C3⋊S3)⋊4C2, SmallGroup(288,944)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×D12⋊S3
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — C2×S3×Dic3 — C2×D12⋊S3
Lower central
C32 — C3×C6 — C2×D12⋊S3
Upper central
C1 — C22 — C2×C4

Generators and relations for C2×D12⋊S3
 G = < a,b,c,d,e | a2=b12=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ece=b10c, ede=d-1 >

Subgroups: 1234 in 355 conjugacy classes, 116 normal (24 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C4○D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, D4⋊2S3, Q8⋊3S3, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C2×D4⋊2S3, C2×Q8⋊3S3, D12⋊S3, C2×S3×Dic3, C2×C3⋊D12, C6×Dic6, C6×D12, C2×C4×C3⋊S3, C2×D12⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, D4⋊2S3, Q8⋊3S3, S3×C23, C2×S32, C2×D4⋊2S3, C2×Q8⋊3S3, D12⋊S3, C22×S32, C2×D12⋊S3

Smallest permutation representation of C2×D12⋊S3
►On 48 points
Generators in S48
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 48)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222222233344444444446···6666666612···1212121212
size11116666181822422666699992···2444121212124···412121212

48 irreducible representations

dim111111122222222444444
type+++++++++++++++-+++
imageC1C2C2C2C2C2C2S3S3D6D6D6D6D6C4○D4S32D4⋊2S3Q8⋊3S3C2×S32C2×S32D12⋊S3
kernelC2×D12⋊S3D12⋊S3C2×S3×Dic3C2×C3⋊D12C6×Dic6C6×D12C2×C4×C3⋊S3C2×Dic6C2×D12Dic6D12C2×Dic3C2×C12C22×S3C3×C6C2×C4C6C6C4C22C2
# reps182211111442224122214

Matrix representation of C2×D12⋊S3 ►in GL6(𝔽13)

1200000
0120000
001000
000100
0000120
0000012
,
010000
1210000
008000
005500
000010
000001
,
1210000
010000
00121100
000100
000010
000001
,
100000
010000
001000
000100
00001212
000010
,
0120000
1200000
001000
00121200
0000120
000011

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;
 

C2×D12⋊S3 in GAP, Magma, Sage, TeX 

C_2\times D_{12}\rtimes S_3
 
% in TeX
 
G:=Group("C2xD12:S3");
 
// GroupNames label
 
G:=SmallGroup(288,944);
 
// by ID
 
G=gap.SmallGroup(288,944);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,675,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
 
// generators/relations
 

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