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## G = C2×S3×Dic3order 144 = 24·32

### Direct product of C2, S3 and Dic3

Series: Derived Chief Lower central Upper central

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

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, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, C3⋊Dic3, S3×C6, C62, S3×C2×C4, C22×Dic3, S3×Dic3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C2×S3×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, S32, S3×C2×C4, C22×Dic3, S3×Dic3, 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 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 size 1 1 1 1 3 3 3 3 2 2 4 3 3 3 3 9 9 9 9 2 ··· 2 4 4 4 6 6 6 6 6 6 6 6

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C4 S3 S3 D6 Dic3 D6 D6 C4×S3 S32 S3×Dic3 C2×S32 kernel C2×S3×Dic3 S3×Dic3 C6×Dic3 C2×C3⋊Dic3 S3×C2×C6 S3×C6 C2×Dic3 C22×S3 Dic3 D6 D6 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 1 2 4 2 2 4 1 2 1

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

 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 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 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 12 0 0 0 0 1 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

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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