Copied to
clipboard

G = C3×C12.58D6order 432 = 24·33

Direct product of C3 and C12.58D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12.58D6, C62.24C12, C3317M4(2), C62.24Dic3, (C6×C12).30C6, (C6×C12).40S3, (C3×C62).9C4, C12.104(S3×C6), (C3×C12).15C12, C324C816C6, (C3×C12).226D6, (C32×C12).9C4, C12.7(C3×Dic3), C6.25(C6×Dic3), (C3×C12).20Dic3, C12.11(C3⋊Dic3), C3211(C3×M4(2)), (C32×C12).94C22, C3210(C4.Dic3), (C3×C6×C12).8C2, C4.(C3×C3⋊Dic3), C4.15(C6×C3⋊S3), C12.97(C2×C3⋊S3), C2.3(C6×C3⋊Dic3), C32(C3×C4.Dic3), (C2×C12).28(C3×S3), (C3×C12).99(C2×C6), (C3×C6).62(C2×C12), C22.(C3×C3⋊Dic3), C6.21(C2×C3⋊Dic3), (C2×C12).25(C3⋊S3), (C3×C324C8)⋊20C2, (C32×C6).69(C2×C4), (C2×C6).6(C3⋊Dic3), (C2×C6).25(C3×Dic3), (C3×C6).67(C2×Dic3), (C2×C4).2(C3×C3⋊S3), SmallGroup(432,486)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3×C12.58D6
C1C3C32C3×C6C3×C12C32×C12C3×C324C8 — C3×C12.58D6
C32C3×C6 — C3×C12.58D6
C1C12C2×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 2A2B3A3B3C···3N4A4B4C6A6B6C···6AN8A8B8C8D12A12B12C12D12E···12BB24A···24H
order122333···3444666···688881212121212···1224···24
size112112···2112112···21818181811112···218···18

126 irreducible representations

dim1111111111222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6Dic3M4(2)C3×S3C3×Dic3S3×C6C3×Dic3C4.Dic3C3×M4(2)C3×C4.Dic3
kernelC3×C12.58D6C3×C324C8C3×C6×C12C12.58D6C32×C12C3×C62C324C8C6×C12C3×C12C62C6×C12C3×C12C3×C12C62C33C2×C12C12C12C2×C6C32C32C3
# reps121222424444442888816432

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

8000
0800
0080
0008
,
65000
20900
00270
00027
,
8000
536400
0080
0009
,
201700
585300
00064
0030
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

׿
×
𝔽