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## G = C32×C4○D12order 432 = 24·33

### Direct product of C32 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×C4○D12
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C3×C12 — C32×C4○D12
 Lower central C3 — C6 — C32×C4○D12
 Upper central C1 — C3×C12 — C6×C12

Generators and relations for C32×C4○D12
G = < a,b,c,d,e | a3=b3=c4=e2=1, d6=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >

Subgroups: 584 in 304 conjugacy classes, 138 normal (30 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, C62, C62, C4○D12, C3×C4○D4, S3×C32, C32×C6, C32×C6, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, C6×C12, D4×C32, Q8×C32, C32×Dic3, C32×C12, S3×C3×C6, C3×C62, C3×C4○D12, C32×C4○D4, C32×Dic6, S3×C3×C12, C32×D12, C32×C3⋊D4, C3×C6×C12, C32×C4○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, C32, D6, C2×C6, C4○D4, C3×S3, C3×C6, C22×S3, C22×C6, S3×C6, C62, C4○D12, C3×C4○D4, S3×C32, S3×C2×C6, C2×C62, S3×C3×C6, C3×C4○D12, C32×C4○D4, S3×C62, C32×C4○D12

Smallest permutation representation of C32×C4○D12
On 72 points
Generators in S72
(1 32 60)(2 33 49)(3 34 50)(4 35 51)(5 36 52)(6 25 53)(7 26 54)(8 27 55)(9 28 56)(10 29 57)(11 30 58)(12 31 59)(13 61 46)(14 62 47)(15 63 48)(16 64 37)(17 65 38)(18 66 39)(19 67 40)(20 68 41)(21 69 42)(22 70 43)(23 71 44)(24 72 45)
(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 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 64 67 70)(62 65 68 71)(63 66 69 72)
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 48)(10 47)(11 46)(12 45)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 72)(60 71)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 4E 6A ··· 6H 6I ··· 6AQ 6AR ··· 6BG 12A ··· 12P 12Q ··· 12BH 12BI ··· 12BX order 1 2 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 2 6 6 1 ··· 1 2 ··· 2 1 1 2 6 6 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 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 C4○D12 C3×C4○D4 C3×C4○D12 kernel C32×C4○D12 C32×Dic6 S3×C3×C12 C32×D12 C32×C3⋊D4 C3×C6×C12 C3×C4○D12 C3×Dic6 S3×C12 C3×D12 C3×C3⋊D4 C6×C12 C6×C12 C3×C12 C62 C33 C2×C12 C12 C2×C6 C32 C32 C3 # reps 1 1 2 1 2 1 8 8 16 8 16 8 1 2 1 2 8 16 8 4 16 32

Matrix representation of C32×C4○D12 in GL4(𝔽13) generated by

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

C32×C4○D12 in GAP, Magma, Sage, TeX

C_3^2\times C_4\circ D_{12}
% in TeX

G:=Group("C3^2xC4oD12");
// GroupNames label

G:=SmallGroup(432,703);
// by ID

G=gap.SmallGroup(432,703);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,512,1598,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^4=e^2=1,d^6=c^2,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,c*d=d*c,c*e=e*c,e*d*e=c^2*d^5>;
// generators/relations

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