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G = C2×S3×Dic3order 144 = 24·32

Direct product of C2, S3 and Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×S3×Dic3, D6.9D6, C62.6C22, C63(C4×S3), (S3×C6)⋊3C4, C22.8S32, C61(C2×Dic3), (C2×C6).13D6, (C6×Dic3)⋊7C2, C323(C22×C4), (C22×S3).2S3, (S3×C6).9C22, C6.10(C22×S3), (C3×C6).10C23, C3⋊Dic34C22, C31(C22×Dic3), (C3×Dic3)⋊6C22, C34(S3×C2×C4), C2.2(C2×S32), (C3×C6)⋊2(C2×C4), (S3×C2×C6).3C2, (C3×S3)⋊2(C2×C4), (C2×C3⋊Dic3)⋊4C2, SmallGroup(144,146)

Series: Derived Chief Lower central Upper central

C1C32 — C2×S3×Dic3
C1C3C32C3×C6S3×C6S3×Dic3 — C2×S3×Dic3
C32 — C2×S3×Dic3
C1C22

Generators and relations for C2×S3×Dic3
 G = < a,b,c,d,e | a2=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 288 in 116 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×4], C22, C22 [×6], S3 [×4], C6 [×2], C6 [×4], C6 [×7], C2×C4 [×6], C23, C32, Dic3 [×2], Dic3 [×6], C12 [×2], D6 [×6], C2×C6 [×2], C2×C6 [×7], C22×C4, C3×S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×4], C2×Dic3, C2×Dic3 [×7], C2×C12, C22×S3, C22×C6, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×6], C62, S3×C2×C4, C22×Dic3, S3×Dic3 [×4], C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C2×S3×Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], S32, S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, C2×S3×Dic3

Smallest permutation representation of C2×S3×Dic3
On 48 points
Generators in S48
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 40)(20 41)(21 42)(22 37)(23 38)(24 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 20 4 23)(2 19 5 22)(3 24 6 21)(7 14 10 17)(8 13 11 16)(9 18 12 15)(25 44 28 47)(26 43 29 46)(27 48 30 45)(31 38 34 41)(32 37 35 40)(33 42 36 39)

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

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

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

C2×S3×Dic3 is a maximal subgroup of
C62.47C23  C62.48C23  C62.49C23  Dic34D12  C62.51C23  C62.54C23  C62.55C23  Dic3⋊D12  D61Dic6  D6.D12  D6.9D12  D62Dic6  D63Dic6  D12⋊Dic3  D64Dic6  C62.72C23  C62.111C23  C62.112C23  C62.113C23  C62.115C23  S32×C2×C4
C2×S3×Dic3 is a maximal quotient of
D12.2Dic3  D12.Dic3  C62.11C23  C62.13C23  C62.25C23  D12⋊Dic3  C62.97C23  C62.115C23

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K6L6M12A12B12C12D
order12222222333444444446···6666666612121212
size11113333224333399992···244466666666

36 irreducible representations

dim1111112222222444
type++++++++-+++-+
imageC1C2C2C2C2C4S3S3D6Dic3D6D6C4×S3S32S3×Dic3C2×S32
kernelC2×S3×Dic3S3×Dic3C6×Dic3C2×C3⋊Dic3S3×C2×C6S3×C6C2×Dic3C22×S3Dic3D6D6C2×C6C6C22C2C2
# reps1411181124224121

Matrix representation of C2×S3×Dic3 in GL6(𝔽13)

100000
010000
0012000
0001200
000010
000001
,
1210000
1200000
0012100
0012000
000010
000001
,
0120000
1200000
0001200
0012000
000010
000001
,
1200000
0120000
001000
000100
0000012
0000112
,
500000
050000
001000
000100
000001
000010

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

C2×S3×Dic3 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_3
% in TeX

G:=Group("C2xS3xDic3");
// GroupNames label

G:=SmallGroup(144,146);
// by ID

G=gap.SmallGroup(144,146);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,490,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^6=1,e^2=d^3,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^-1=d^-1>;
// generators/relations

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