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G = C3×D12.S3order 432 = 24·33

Direct product of C3 and D12.S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×D12.S3
G = < a,b,c,d,e | a3=b12=c2=d3=1, e2=b9, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >

Subgroups: 440 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⋊Dic3, C3×C12, C3×C12, S3×C6, C62, C24⋊C2, D4.S3, C3×SD16, S3×C32, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C3×C24, C3×Dic6, C3×D12, C3×D12, C324Q8, D4×C32, C3×C3⋊Dic3, C32×C12, S3×C3×C6, D12.S3, C3×C24⋊C2, C3×D4.S3, C32×C3⋊C8, C32×D12, C3×C324Q8, C3×D12.S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, D12, C3⋊D4, C3×D4, S32, S3×C6, C24⋊C2, D4.S3, C3×SD16, C3⋊D12, C3×D12, C3×C3⋊D4, C3×S32, D12.S3, C3×C24⋊C2, C3×D4.S3, C3×C3⋊D12, C3×D12.S3

Smallest permutation representation of C3×D12.S3
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)
(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 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 19 10 16 7 13 4 22)(2 20 11 17 8 14 5 23)(3 21 12 18 9 15 6 24)(25 41 34 38 31 47 28 44)(26 42 35 39 32 48 29 45)(27 43 36 40 33 37 30 46)

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

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

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

72 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 ··· 12H 12I ··· 12Q 12R 12S 24A ··· 24P 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 size 1 1 12 1 1 2 ··· 2 4 4 4 2 36 1 1 2 ··· 2 4 4 4 12 ··· 12 6 6 2 ··· 2 4 ··· 4 36 36 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 D12 C3⋊D4 C3×D4 S3×C6 C24⋊C2 C3×SD16 C3×D12 C3×C3⋊D4 C3×C24⋊C2 S32 D4.S3 C3⋊D12 C3×S32 D12.S3 C3×D4.S3 C3×C3⋊D12 C3×D12.S3 kernel C3×D12.S3 C32×C3⋊C8 C32×D12 C3×C32⋊4Q8 D12.S3 C3×C3⋊C8 C3×D12 C32⋊4Q8 C3×C3⋊C8 C3×D12 C32×C6 C3×C12 C33 C3⋊C8 D12 C3×C6 C3×C6 C3×C6 C12 C32 C32 C6 C6 C3 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 1 1 1 2 2 2 2 4

Matrix representation of C3×D12.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 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 9 0 0 0 0 46 68 0 0 0 0 0 0 0 72 0 0 0 0 72 0 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 61 0 0 0 0 6 61 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

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],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,46,0,0,0,0,9,68,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,6,0,0,0,0,61,61,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×D12.S3 in GAP, Magma, Sage, TeX

C_3\times D_{12}.S_3
% in TeX

G:=Group("C3xD12.S3");
// GroupNames label

G:=SmallGroup(432,421);
// by ID

G=gap.SmallGroup(432,421);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^3=1,e^2=b^9,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,e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations

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