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## G = C32×C6.D4order 432 = 24·33

### Direct product of C32 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C3×C62 — Dic3×C3×C6 — C32×C6.D4
 Lower central C3 — C6 — C32×C6.D4
 Upper central C1 — C62 — C2×C62

Generators and relations for C32×C6.D4
G = < a,b,c,d,e | a3=b3=c6=d4=1, e2=c3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c3d-1 >

Subgroups: 616 in 332 conjugacy classes, 114 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C33, C3×Dic3, C3×C12, C62, C62, C62, C6.D4, C3×C22⋊C4, C32×C6, C32×C6, C32×C6, C6×Dic3, C6×C12, C2×C62, C2×C62, C2×C62, C32×Dic3, C3×C62, C3×C62, C3×C62, C3×C6.D4, C32×C22⋊C4, Dic3×C3×C6, C63, C32×C6.D4
Quotients:

Smallest permutation representation of C32×C6.D4
On 72 points
Generators in S72
(1 27 40)(2 28 41)(3 29 42)(4 30 37)(5 25 38)(6 26 39)(7 54 64)(8 49 65)(9 50 66)(10 51 61)(11 52 62)(12 53 63)(13 24 32)(14 19 33)(15 20 34)(16 21 35)(17 22 36)(18 23 31)(43 59 69)(44 60 70)(45 55 71)(46 56 72)(47 57 67)(48 58 68)
(1 42 25)(2 37 26)(3 38 27)(4 39 28)(5 40 29)(6 41 30)(7 62 50)(8 63 51)(9 64 52)(10 65 53)(11 66 54)(12 61 49)(13 34 22)(14 35 23)(15 36 24)(16 31 19)(17 32 20)(18 33 21)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)
(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 43 14 50)(2 48 15 49)(3 47 16 54)(4 46 17 53)(5 45 18 52)(6 44 13 51)(7 42 67 35)(8 41 68 34)(9 40 69 33)(10 39 70 32)(11 38 71 31)(12 37 72 36)(19 66 27 59)(20 65 28 58)(21 64 29 57)(22 63 30 56)(23 62 25 55)(24 61 26 60)
(1 46 4 43)(2 45 5 48)(3 44 6 47)(7 35 10 32)(8 34 11 31)(9 33 12 36)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 63 22 66)(20 62 23 65)(21 61 24 64)(25 58 28 55)(26 57 29 60)(27 56 30 59)(37 69 40 72)(38 68 41 71)(39 67 42 70)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 6A ··· 6X 6Y ··· 6CY 12A ··· 12AF order 1 2 2 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Dic3 D6 C3×S3 C3⋊D4 C3×D4 C3×Dic3 S3×C6 C3×C3⋊D4 kernel C32×C6.D4 Dic3×C3×C6 C63 C3×C6.D4 C3×C62 C6×Dic3 C2×C62 C62 C2×C62 C32×C6 C62 C62 C22×C6 C3×C6 C3×C6 C2×C6 C2×C6 C6 # reps 1 2 1 8 4 16 8 32 1 2 2 1 8 4 16 16 8 32

Matrix representation of C32×C6.D4 in GL3(𝔽13) generated by

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

C32×C6.D4 in GAP, Magma, Sage, TeX

C_3^2\times C_6.D_4
% in TeX

G:=Group("C3^2xC6.D4");
// GroupNames label

G:=SmallGroup(432,479);
// by ID

G=gap.SmallGroup(432,479);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,1037,14118]);
// Polycyclic

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

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