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## G = C3×C6.D12order 432 = 24·33

### Direct product of C3 and C6.D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C6.D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — Dic3×C3×C6 — C3×C6.D12
 Lower central C32 — C3×C6 — C3×C6.D12
 Upper central C1 — C2×C6

Generators and relations for C3×C6.D12
G = < a,b,c,d | a3=b6=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 800 in 210 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C3×C22⋊C4, C3×C3⋊S3, C32×C6, C32×C6, C6×Dic3, C6×Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C6.D12, C3×D6⋊C4, Dic3×C3×C6, C2×C6×C3⋊S3, C3×C6.D12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, S32, S3×C6, D6⋊C4, C3×C22⋊C4, C6.D6, C3⋊D12, S3×C12, C3×D12, C3×C3⋊D4, C3×S32, C6.D12, C3×D6⋊C4, C3×C6.D6, C3×C3⋊D12, C3×C6.D12

Smallest permutation representation of C3×C6.D12
On 48 points
Generators in S48
(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 43 9 39 5 47)(2 48 6 40 10 44)(3 45 11 41 7 37)(4 38 8 42 12 46)(13 34 21 30 17 26)(14 27 18 31 22 35)(15 36 23 32 19 28)(16 29 20 33 24 25)
(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 19)(2 35)(3 17)(4 33)(5 15)(6 31)(7 13)(8 29)(9 23)(10 27)(11 21)(12 25)(14 44)(16 42)(18 40)(20 38)(22 48)(24 46)(26 37)(28 47)(30 45)(32 43)(34 41)(36 39)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 6AH 6AI 6AJ 6AK 12A ··· 12AF order 1 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 1 1 18 18 1 1 2 ··· 2 4 4 4 6 6 6 6 1 ··· 1 2 ··· 2 4 ··· 4 18 18 18 18 6 ··· 6

90 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 type + + + + + + + + + + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 D6 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 S32 C6.D6 C3⋊D12 C3×S32 C3×C6.D6 C3×C3⋊D12 kernel C3×C6.D12 Dic3×C3×C6 C2×C6×C3⋊S3 C6.D12 C6×C3⋊S3 C6×Dic3 C22×C3⋊S3 C2×C3⋊S3 C6×Dic3 C32×C6 C62 C2×Dic3 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 2 2 2 4 4 4 4 4 4 8 8 8 1 1 2 2 2 4

Matrix representation of C3×C6.D12 in GL8(𝔽13)

 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3×C6.D12 in GAP, Magma, Sage, TeX

C_3\times C_6.D_{12}
% in TeX

G:=Group("C3xC6.D12");
// GroupNames label

G:=SmallGroup(432,427);
// by ID

G=gap.SmallGroup(432,427);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,260,2028,14118]);
// Polycyclic

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

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