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G = C2×S3×D12order 288 = 25·32

Direct product of C2, S3 and D12

direct product, metabelian, supersoluble, monomial

Aliases: C2×S3×D12, C62.135C23, C61(S3×D4), C61(C2×D12), (C4×S3)⋊14D6, (S3×C6)⋊14D4, (C2×C12)⋊21D6, (C6×D12)⋊15C2, (S3×C6)⋊1C23, (C6×C12)⋊3C22, (C3×C12)⋊2C23, C124(C22×S3), (C22×S3)⋊9D6, D61(C22×S3), C31(C22×D12), C322(C22×D4), (C2×Dic3)⋊19D6, C6.10(S3×C23), (C3×C6).10C24, (S3×C12)⋊18C22, (C3×D12)⋊27C22, C3⋊D128C22, (C3×Dic3)⋊4C23, Dic34(C22×S3), C12⋊S321C22, (C6×Dic3)⋊26C22, C42(C2×S32), (C2×C4)⋊6S32, C31(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

C1C3×C6 — C2×S3×D12
C1C3C32C3×C6S3×C6C2×S32C22×S32 — C2×S3×D12
C32C3×C6 — C2×S3×D12
C1C22C2×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 [×2], C2 [×12], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×38], S3 [×4], S3 [×16], C6 [×2], C6 [×4], C6 [×11], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×10], D6 [×60], C2×C6 [×2], C2×C6 [×15], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3×S3 [×4], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×4], D12 [×4], D12 [×20], C2×Dic3, C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×6], C3×D4 [×4], C22×S3, C22×S3 [×2], C22×S3 [×38], C22×C6 [×3], C22×D4, C3×Dic3 [×2], C3×C12 [×2], S32 [×16], S3×C6 [×10], S3×C6 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, S3×C2×C4, C2×D12, C2×D12 [×13], S3×D4 [×8], C2×C3⋊D4 [×2], C22×C12, C6×D4, S3×C23 [×4], C3⋊D12 [×8], S3×C12 [×4], C3×D12 [×4], C6×Dic3, C12⋊S3 [×4], C6×C12, C2×S32 [×8], C2×S32 [×8], S3×C2×C6, S3×C2×C6 [×2], C22×C3⋊S3 [×2], C22×D12, C2×S3×D4, S3×D12 [×8], C2×C3⋊D12 [×2], S3×C2×C12, C6×D12, C2×C12⋊S3, C22×S32 [×2], C2×S3×D12
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, D12 [×4], C22×S3 [×14], C22×D4, S32, C2×D12 [×6], S3×D4 [×2], S3×C23 [×2], C2×S32 [×3], C22×D12, C2×S3×D4, S3×D12 [×2], C22×S32, C2×S3×D12

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

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

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

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

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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