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## G = C3×C6×C3⋊D4order 432 = 24·33

### Direct product of C3×C6 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C6×C3⋊D4
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C62 — C3×C6×C3⋊D4
 Lower central C3 — C6 — C3×C6×C3⋊D4
 Upper central C1 — C62 — C2×C62

Generators and relations for C3×C6×C3⋊D4
G = < a,b,c,d,e | a3=b6=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 904 in 452 conjugacy classes, 162 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22×C6, C33, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, S3×C32, C32×C6, C32×C6, C32×C6, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, C2×C62, C2×C62, C2×C62, C32×Dic3, S3×C3×C6, S3×C3×C6, C3×C62, C3×C62, C3×C62, C6×C3⋊D4, D4×C3×C6, Dic3×C3×C6, C32×C3⋊D4, S3×C62, C63, C3×C6×C3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, C32, D6, C2×C6, C2×D4, C3×S3, C3×C6, C3⋊D4, C3×D4, C22×S3, C22×C6, S3×C6, C62, C2×C3⋊D4, C6×D4, S3×C32, C3×C3⋊D4, D4×C32, S3×C2×C6, C2×C62, S3×C3×C6, C6×C3⋊D4, D4×C3×C6, C32×C3⋊D4, S3×C62, C3×C6×C3⋊D4

Smallest permutation representation of C3×C6×C3⋊D4
On 72 points
Generators in S72
(1 57 49)(2 58 50)(3 59 51)(4 60 52)(5 55 53)(6 56 54)(7 23 33)(8 24 34)(9 19 35)(10 20 36)(11 21 31)(12 22 32)(13 25 68)(14 26 69)(15 27 70)(16 28 71)(17 29 72)(18 30 67)(37 44 61)(38 45 62)(39 46 63)(40 47 64)(41 48 65)(42 43 66)
(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 53 59)(2 54 60)(3 49 55)(4 50 56)(5 51 57)(6 52 58)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(13 27 72)(14 28 67)(15 29 68)(16 30 69)(17 25 70)(18 26 71)(37 65 46)(38 66 47)(39 61 48)(40 62 43)(41 63 44)(42 64 45)
(1 15 65 33)(2 16 66 34)(3 17 61 35)(4 18 62 36)(5 13 63 31)(6 14 64 32)(7 57 27 41)(8 58 28 42)(9 59 29 37)(10 60 30 38)(11 55 25 39)(12 56 26 40)(19 51 72 44)(20 52 67 45)(21 53 68 46)(22 54 69 47)(23 49 70 48)(24 50 71 43)
(1 36)(2 31)(3 32)(4 33)(5 34)(6 35)(7 60)(8 55)(9 56)(10 57)(11 58)(12 59)(13 66)(14 61)(15 62)(16 63)(17 64)(18 65)(19 54)(20 49)(21 50)(22 51)(23 52)(24 53)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)(43 68)(44 69)(45 70)(46 71)(47 72)(48 67)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6X 6Y ··· 6CY 6CZ ··· 6DO 12A ··· 12P order 1 2 2 2 2 2 2 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 6 6 1 ··· 1 2 ··· 2 6 6 1 ··· 1 2 ··· 2 6 ··· 6 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 kernel C3×C6×C3⋊D4 Dic3×C3×C6 C32×C3⋊D4 S3×C62 C63 C6×C3⋊D4 C6×Dic3 C3×C3⋊D4 S3×C2×C6 C2×C62 C2×C62 C32×C6 C62 C22×C6 C3×C6 C3×C6 C2×C6 C6 # reps 1 1 4 1 1 8 8 32 8 8 1 2 3 8 4 16 24 32

Matrix representation of C3×C6×C3⋊D4 in GL4(𝔽13) generated by

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

C3×C6×C3⋊D4 in GAP, Magma, Sage, TeX

C_3\times C_6\times C_3\rtimes D_4
% in TeX

G:=Group("C3xC6xC3:D4");
// GroupNames label

G:=SmallGroup(432,709);
// by ID

G=gap.SmallGroup(432,709);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1598,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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