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## G = C3×C12.D6order 432 = 24·33

### Direct product of C3 and C12.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C12.D6
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×C3⋊S3 — C12×C3⋊S3 — C3×C12.D6
 Lower central C32 — C3×C6 — C3×C12.D6
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×C12.D6
G = < a,b,c,d | a3=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 852 in 304 conjugacy classes, 94 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, D42S3, C3×C4○D4, C3×C3⋊S3, C32×C6, C32×C6, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, D4×C32, D4×C32, C3×C3⋊Dic3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×C62, C3×D42S3, C12.D6, C3×C324Q8, C12×C3⋊S3, C6×C3⋊Dic3, C3×C327D4, D4×C33, C3×C12.D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, D42S3, C3×C4○D4, C3×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C3×D42S3, C12.D6, C2×C6×C3⋊S3, C3×C12.D6

Smallest permutation representation of C3×C12.D6
On 72 points
Generators in S72
(1 62 58)(2 63 59)(3 64 60)(4 65 49)(5 66 50)(6 67 51)(7 68 52)(8 69 53)(9 70 54)(10 71 55)(11 72 56)(12 61 57)(13 43 35)(14 44 36)(15 45 25)(16 46 26)(17 47 27)(18 48 28)(19 37 29)(20 38 30)(21 39 31)(22 40 32)(23 41 33)(24 42 34)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 54 66)(2 49 67 8 55 61)(3 56 68)(4 51 69 10 57 63)(5 58 70)(6 53 71 12 59 65)(7 60 72)(9 50 62)(11 52 64)(13 39 27)(14 46 28 20 40 34)(15 41 29)(16 48 30 22 42 36)(17 43 31)(18 38 32 24 44 26)(19 45 33)(21 47 35)(23 37 25)
(1 38 7 44)(2 37 8 43)(3 48 9 42)(4 47 10 41)(5 46 11 40)(6 45 12 39)(13 59 19 53)(14 58 20 52)(15 57 21 51)(16 56 22 50)(17 55 23 49)(18 54 24 60)(25 61 31 67)(26 72 32 66)(27 71 33 65)(28 70 34 64)(29 69 35 63)(30 68 36 62)

G:=sub<Sym(72)| (1,62,58)(2,63,59)(3,64,60)(4,65,49)(5,66,50)(6,67,51)(7,68,52)(8,69,53)(9,70,54)(10,71,55)(11,72,56)(12,61,57)(13,43,35)(14,44,36)(15,45,25)(16,46,26)(17,47,27)(18,48,28)(19,37,29)(20,38,30)(21,39,31)(22,40,32)(23,41,33)(24,42,34), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,54,66)(2,49,67,8,55,61)(3,56,68)(4,51,69,10,57,63)(5,58,70)(6,53,71,12,59,65)(7,60,72)(9,50,62)(11,52,64)(13,39,27)(14,46,28,20,40,34)(15,41,29)(16,48,30,22,42,36)(17,43,31)(18,38,32,24,44,26)(19,45,33)(21,47,35)(23,37,25), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,59,19,53)(14,58,20,52)(15,57,21,51)(16,56,22,50)(17,55,23,49)(18,54,24,60)(25,61,31,67)(26,72,32,66)(27,71,33,65)(28,70,34,64)(29,69,35,63)(30,68,36,62)>;

G:=Group( (1,62,58)(2,63,59)(3,64,60)(4,65,49)(5,66,50)(6,67,51)(7,68,52)(8,69,53)(9,70,54)(10,71,55)(11,72,56)(12,61,57)(13,43,35)(14,44,36)(15,45,25)(16,46,26)(17,47,27)(18,48,28)(19,37,29)(20,38,30)(21,39,31)(22,40,32)(23,41,33)(24,42,34), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,54,66)(2,49,67,8,55,61)(3,56,68)(4,51,69,10,57,63)(5,58,70)(6,53,71,12,59,65)(7,60,72)(9,50,62)(11,52,64)(13,39,27)(14,46,28,20,40,34)(15,41,29)(16,48,30,22,42,36)(17,43,31)(18,38,32,24,44,26)(19,45,33)(21,47,35)(23,37,25), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,59,19,53)(14,58,20,52)(15,57,21,51)(16,56,22,50)(17,55,23,49)(18,54,24,60)(25,61,31,67)(26,72,32,66)(27,71,33,65)(28,70,34,64)(29,69,35,63)(30,68,36,62) );

G=PermutationGroup([[(1,62,58),(2,63,59),(3,64,60),(4,65,49),(5,66,50),(6,67,51),(7,68,52),(8,69,53),(9,70,54),(10,71,55),(11,72,56),(12,61,57),(13,43,35),(14,44,36),(15,45,25),(16,46,26),(17,47,27),(18,48,28),(19,37,29),(20,38,30),(21,39,31),(22,40,32),(23,41,33),(24,42,34)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,54,66),(2,49,67,8,55,61),(3,56,68),(4,51,69,10,57,63),(5,58,70),(6,53,71,12,59,65),(7,60,72),(9,50,62),(11,52,64),(13,39,27),(14,46,28,20,40,34),(15,41,29),(16,48,30,22,42,36),(17,43,31),(18,38,32,24,44,26),(19,45,33),(21,47,35),(23,37,25)], [(1,38,7,44),(2,37,8,43),(3,48,9,42),(4,47,10,41),(5,46,11,40),(6,45,12,39),(13,59,19,53),(14,58,20,52),(15,57,21,51),(16,56,22,50),(17,55,23,49),(18,54,24,60),(25,61,31,67),(26,72,32,66),(27,71,33,65),(28,70,34,64),(29,69,35,63),(30,68,36,62)]])

90 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3N 4A 4B 4C 4D 4E 6A 6B 6C ··· 6R 6S ··· 6AP 6AQ 6AR 12A 12B 12C ··· 12N 12O 12P 12Q 12R 12S 12T 12U 12V order 1 2 2 2 2 3 3 3 ··· 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 2 2 18 1 1 2 ··· 2 2 9 9 18 18 1 1 2 ··· 2 4 ··· 4 18 18 2 2 4 ··· 4 9 9 9 9 18 18 18 18

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C3×C4○D4 D4⋊2S3 C3×D4⋊2S3 kernel C3×C12.D6 C3×C32⋊4Q8 C12×C3⋊S3 C6×C3⋊Dic3 C3×C32⋊7D4 D4×C33 C12.D6 C32⋊4Q8 C4×C3⋊S3 C2×C3⋊Dic3 C32⋊7D4 D4×C32 D4×C32 C3×C12 C62 C33 C3×D4 C12 C2×C6 C32 C32 C3 # reps 1 1 1 2 2 1 2 2 2 4 4 2 4 4 8 2 8 8 16 4 4 8

Matrix representation of C3×C12.D6 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 0 0 0 0 0 1 4 0 0 0 0 0 0 9 1 0 0 0 0 0 3 0 0 0 0 0 0 12 5 0 0 0 0 10 1
,
 3 0 0 0 0 0 12 9 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 3 12
,
 6 3 0 0 0 0 10 7 0 0 0 0 0 0 10 1 0 0 0 0 5 3 0 0 0 0 0 0 5 1 0 0 0 0 0 8

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,1,0,0,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,1,3,0,0,0,0,0,0,12,10,0,0,0,0,5,1],[3,12,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,12],[6,10,0,0,0,0,3,7,0,0,0,0,0,0,10,5,0,0,0,0,1,3,0,0,0,0,0,0,5,0,0,0,0,0,1,8] >;

C3×C12.D6 in GAP, Magma, Sage, TeX

C_3\times C_{12}.D_6
% in TeX

G:=Group("C3xC12.D6");
// GroupNames label

G:=SmallGroup(432,715);
// by ID

G=gap.SmallGroup(432,715);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,4037,14118]);
// Polycyclic

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

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