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## G = C3×C12.58D6order 432 = 24·33

### Direct product of C3 and C12.58D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C12.58D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C32⋊4C8 — C3×C12.58D6
 Lower central C32 — C3×C6 — C3×C12.58D6
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C12.58D6
G = < a,b,c,d | a3=b12=c6=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 356 in 184 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C3, C3 [×4], C3 [×4], C4 [×2], C22, C6, C6 [×4], C6 [×17], C8 [×2], C2×C4, C32, C32 [×4], C32 [×4], C12 [×2], C12 [×8], C12 [×8], C2×C6, C2×C6 [×4], C2×C6 [×4], M4(2), C3×C6, C3×C6 [×4], C3×C6 [×17], C3⋊C8 [×8], C24 [×2], C2×C12, C2×C12 [×4], C2×C12 [×4], C33, C3×C12 [×2], C3×C12 [×8], C3×C12 [×8], C62, C62 [×4], C62 [×4], C4.Dic3 [×4], C3×M4(2), C32×C6, C32×C6, C3×C3⋊C8 [×8], C324C8 [×2], C6×C12, C6×C12 [×4], C6×C12 [×4], C32×C12 [×2], C3×C62, C3×C4.Dic3 [×4], C12.58D6, C3×C324C8 [×2], C3×C6×C12, C3×C12.58D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3 [×4], C6 [×3], C2×C4, Dic3 [×8], C12 [×2], D6 [×4], C2×C6, M4(2), C3×S3 [×4], C3⋊S3, C2×Dic3 [×4], C2×C12, C3×Dic3 [×8], C3⋊Dic3 [×2], S3×C6 [×4], C2×C3⋊S3, C4.Dic3 [×4], C3×M4(2), C3×C3⋊S3, C6×Dic3 [×4], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C6×C3⋊S3, C3×C4.Dic3 [×4], C12.58D6, C6×C3⋊Dic3, C3×C12.58D6

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

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

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

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

126 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3N 4A 4B 4C 6A 6B 6C ··· 6AN 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12BB 24A ··· 24H order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 1 1 2 ··· 2 1 1 2 1 1 2 ··· 2 18 18 18 18 1 1 1 1 2 ··· 2 18 ··· 18

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×Dic3 S3×C6 C3×Dic3 C4.Dic3 C3×M4(2) C3×C4.Dic3 kernel C3×C12.58D6 C3×C32⋊4C8 C3×C6×C12 C12.58D6 C32×C12 C3×C62 C32⋊4C8 C6×C12 C3×C12 C62 C6×C12 C3×C12 C3×C12 C62 C33 C2×C12 C12 C12 C2×C6 C32 C32 C3 # reps 1 2 1 2 2 2 4 2 4 4 4 4 4 4 2 8 8 8 8 16 4 32

Matrix representation of C3×C12.58D6 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 65 0 0 0 20 9 0 0 0 0 27 0 0 0 0 27
,
 8 0 0 0 53 64 0 0 0 0 8 0 0 0 0 9
,
 20 17 0 0 58 53 0 0 0 0 0 64 0 0 3 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[65,20,0,0,0,9,0,0,0,0,27,0,0,0,0,27],[8,53,0,0,0,64,0,0,0,0,8,0,0,0,0,9],[20,58,0,0,17,53,0,0,0,0,0,3,0,0,64,0] >;

C3×C12.58D6 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{58}D_6
% in TeX

G:=Group("C3xC12.58D6");
// GroupNames label

G:=SmallGroup(432,486);
// by ID

G=gap.SmallGroup(432,486);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^6=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations

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