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## G = C3×Dic6⋊S3order 432 = 24·33

### Direct product of C3 and Dic6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×Dic6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×D12 — C3×Dic6⋊S3
 Lower central C32 — C3×C6 — C3×C12 — C3×Dic6⋊S3
 Upper central C1 — C6 — C12

Generators and relations for C3×Dic6⋊S3
G = < a,b,c,d,e | a3=b12=d3=e2=1, c2=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b7, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 400 in 122 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, SD16, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, D4.S3, Q82S3, C3×SD16, S3×C32, C32×C6, C3×C3⋊C8, C324C8, C3×Dic6, C3×Dic6, C3×D12, C3×D12, D4×C32, Q8×C32, C32×Dic3, C32×C12, S3×C3×C6, Dic6⋊S3, C3×D4.S3, C3×Q82S3, C3×C324C8, C32×Dic6, C32×D12, C3×Dic6⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊D4, C3×D4, S32, S3×C6, D4.S3, Q82S3, C3×SD16, D6⋊S3, C3×C3⋊D4, C3×S32, Dic6⋊S3, C3×D4.S3, C3×Q82S3, C3×D6⋊S3, C3×Dic6⋊S3

Smallest permutation representation of C3×Dic6⋊S3
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 14 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 37 31 43)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)
(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 28)(2 35)(3 30)(4 25)(5 32)(6 27)(7 34)(8 29)(9 36)(10 31)(11 26)(12 33)(13 48)(14 43)(15 38)(16 45)(17 40)(18 47)(19 42)(20 37)(21 44)(22 39)(23 46)(24 41)

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

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

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

63 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6S 8A 8B 12A 12B 12C ··· 12N 12O ··· 12V 24A 24B 24C 24D order 1 2 2 3 3 3 ··· 3 3 3 3 4 4 6 6 6 ··· 6 6 6 6 6 ··· 6 8 8 12 12 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 12 1 1 2 ··· 2 4 4 4 2 12 1 1 2 ··· 2 4 4 4 12 ··· 12 18 18 2 2 4 ··· 4 12 ··· 12 18 18 18 18

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 S3 D4 D6 SD16 C3×S3 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 C3×C3⋊D4 S32 D4.S3 Q8⋊2S3 D6⋊S3 C3×S32 Dic6⋊S3 C3×D4.S3 C3×Q8⋊2S3 C3×D6⋊S3 C3×Dic6⋊S3 kernel C3×Dic6⋊S3 C3×C32⋊4C8 C32×Dic6 C32×D12 Dic6⋊S3 C32⋊4C8 C3×Dic6 C3×D12 C3×Dic6 C3×D12 C32×C6 C3×C12 C33 Dic6 D12 C3×C6 C3×C6 C12 C32 C6 C12 C32 C32 C6 C4 C3 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 2 4 4 8 1 1 1 1 2 2 2 2 2 4

Matrix representation of C3×Dic6⋊S3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 71 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 19 22 0 0 0 0 30 54 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 59 19 0 0 0 0 32 14 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[19,30,0,0,0,0,22,54,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[59,32,0,0,0,0,19,14,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×Dic6⋊S3 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_6\rtimes S_3
% in TeX

G:=Group("C3xDic6:S3");
// GroupNames label

G:=SmallGroup(432,420);
// by ID

G=gap.SmallGroup(432,420);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=d^3=e^2=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^7,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations

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