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G = C2×S3×D12  order 288 = 25·32

Direct product of C2, S3 and D12

direct product, metabelian, supersoluble, monomial

Aliases: C2×S3×D12, C62.135C23, C6⋊1(S3×D4), C6⋊1(C2×D12), (C4×S3)⋊14D6, (S3×C6)⋊14D4, (C2×C12)⋊21D6, (C6×D12)⋊15C2, (S3×C6)⋊1C23, (C6×C12)⋊3C22, (C3×C12)⋊2C23, C12⋊4(C22×S3), (C22×S3)⋊9D6, D6⋊1(C22×S3), C3⋊1(C22×D12), C32⋊2(C22×D4), (C2×Dic3)⋊19D6, C6.10(S3×C23), (C3×C6).10C24, (S3×C12)⋊18C22, (C3×D12)⋊27C22, C3⋊D12⋊8C22, (C3×Dic3)⋊4C23, Dic3⋊4(C22×S3), C12⋊S3⋊21C22, (C6×Dic3)⋊26C22, C4⋊2(C2×S32), (C2×C4)⋊6S32, C3⋊1(C2×S3×D4), (S3×C2×C4)⋊5S3, (S3×C2×C12)⋊9C2, (C3×C6)⋊2(C2×D4), (C2×S32)⋊8C22, (C22×S32)⋊5C2, (C3×S3)⋊1(C2×D4), (S3×C2×C6)⋊6C22, (C2×C3⋊S3)⋊1C23, C22.65(C2×S32), C2.12(C22×S32), (C2×C12⋊S3)⋊18C2, (C2×C3⋊D12)⋊14C2, (C22×C3⋊S3)⋊5C22, (C2×C6).152(C22×S3), SmallGroup(288,951)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C2×S3×D12
 G = < a,b,c,d,e | a2=b3=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2242 in 499 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×D4, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C2×D12, C2×D12, S3×D4, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C3⋊D12, S3×C12, C3×D12, C6×Dic3, C12⋊S3, C6×C12, C2×S32, C2×S32, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, C22×D12, C2×S3×D4, S3×D12, C2×C3⋊D12, S3×C2×C12, C6×D12, C2×C12⋊S3, C22×S32, C2×S3×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, S32, C2×D12, S3×D4, S3×C23, C2×S32, C22×D12, C2×S3×D4, S3×D12, C22×S32, C2×S3×D12

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O3A3B3C4A4B4C4D6A···6F6G6H6I6J6K6L6M6N6O6P6Q12A12B12C12D12E···12J12K12L12M12N
order122222222222222233344446···6666666666661212121212···1212121212
size1111333366661818181822422662···244466661212121222224···46666

54 irreducible representations

dim111111122222222244444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2S3S3D4D6D6D6D6D6D12S32S3×D4C2×S32C2×S32S3×D12
kernelC2×S3×D12S3×D12C2×C3⋊D12S3×C2×C12C6×D12C2×C12⋊S3C22×S32S3×C2×C4C2×D12S3×C6C4×S3D12C2×Dic3C2×C12C22×S3D6C2×C4C6C4C22C2
# reps182111211444123812214

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

100000
010000
001000
000100
0000-10
00000-1
,
010000
-1-10000
001000
000100
000010
000001
,
100000
-1-10000
001000
000100
000010
000001
,
-100000
0-10000
001200
00-1-100
000001
0000-1-1
,
100000
010000
001000
00-1-100
000001
000010

G:=sub<GL(6,Integers())| [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,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C2×S3×D12 in GAP, Magma, Sage, TeX 

C_2\times S_3\times D_{12}
 
% in TeX
 
G:=Group("C2xS3xD12");
 
// GroupNames label
 
G:=SmallGroup(288,951);
 
// by ID
 
G=gap.SmallGroup(288,951);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^12=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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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