Copied to
clipboard

G = C6×C3⋊C16order 288 = 25·32

Direct product of C6 and C3⋊C16

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C6×C3⋊C16, C6⋊C48, C24.5C12, C12.3C24, C24.99D6, C62.6C8, C24.16Dic3, (C3×C6)⋊3C16, C32(C2×C48), C8.21(S3×C6), (C2×C6).5C24, C6.8(C2×C24), C329(C2×C16), C12.11(C3⋊C8), (C2×C24).37S3, (C3×C12).11C8, (C3×C24).10C4, (C6×C12).29C4, C24.34(C2×C6), (C2×C24).21C6, (C6×C24).18C2, C4.9(C6×Dic3), C8.4(C3×Dic3), (C2×C12).16C12, C12.45(C2×C12), (C3×C24).66C22, C12.66(C2×Dic3), (C2×C12).31Dic3, C4.3(C3×C3⋊C8), C2.2(C6×C3⋊C8), C6.19(C2×C3⋊C8), (C2×C6).9(C3⋊C8), (C2×C8).9(C3×S3), C22.2(C3×C3⋊C8), (C3×C6).39(C2×C8), (C2×C4).8(C3×Dic3), (C3×C12).130(C2×C4), SmallGroup(288,245)

Series: Derived Chief Lower central Upper central

C1C3 — C6×C3⋊C16
C1C3C6C12C24C3×C24C3×C3⋊C16 — C6×C3⋊C16
C3 — C6×C3⋊C16
C1C2×C24

Generators and relations for C6×C3⋊C16
 G = < a,b,c | a6=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 90 in 67 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C16 [×2], C2×C8, C3×C6, C3×C6 [×2], C24 [×4], C24 [×2], C2×C12 [×2], C2×C12, C2×C16, C3×C12 [×2], C62, C3⋊C16 [×2], C48 [×2], C2×C24 [×2], C2×C24, C3×C24 [×2], C6×C12, C2×C3⋊C16, C2×C48, C3×C3⋊C16 [×2], C6×C24, C6×C3⋊C16
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C16 [×2], C2×C8, C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, C2×C16, C3×Dic3 [×2], S3×C6, C3⋊C16 [×2], C48 [×2], C2×C3⋊C8, C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C2×C3⋊C16, C2×C48, C3×C3⋊C16 [×2], C6×C3⋊C8, C6×C3⋊C16

Smallest permutation representation of C6×C3⋊C16
On 96 points
Generators in S96
(1 25 45 87 71 64)(2 26 46 88 72 49)(3 27 47 89 73 50)(4 28 48 90 74 51)(5 29 33 91 75 52)(6 30 34 92 76 53)(7 31 35 93 77 54)(8 32 36 94 78 55)(9 17 37 95 79 56)(10 18 38 96 80 57)(11 19 39 81 65 58)(12 20 40 82 66 59)(13 21 41 83 67 60)(14 22 42 84 68 61)(15 23 43 85 69 62)(16 24 44 86 70 63)
(1 45 71)(2 72 46)(3 47 73)(4 74 48)(5 33 75)(6 76 34)(7 35 77)(8 78 36)(9 37 79)(10 80 38)(11 39 65)(12 66 40)(13 41 67)(14 68 42)(15 43 69)(16 70 44)(17 95 56)(18 57 96)(19 81 58)(20 59 82)(21 83 60)(22 61 84)(23 85 62)(24 63 86)(25 87 64)(26 49 88)(27 89 50)(28 51 90)(29 91 52)(30 53 92)(31 93 54)(32 55 94)
(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 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,25,45,87,71,64)(2,26,46,88,72,49)(3,27,47,89,73,50)(4,28,48,90,74,51)(5,29,33,91,75,52)(6,30,34,92,76,53)(7,31,35,93,77,54)(8,32,36,94,78,55)(9,17,37,95,79,56)(10,18,38,96,80,57)(11,19,39,81,65,58)(12,20,40,82,66,59)(13,21,41,83,67,60)(14,22,42,84,68,61)(15,23,43,85,69,62)(16,24,44,86,70,63), (1,45,71)(2,72,46)(3,47,73)(4,74,48)(5,33,75)(6,76,34)(7,35,77)(8,78,36)(9,37,79)(10,80,38)(11,39,65)(12,66,40)(13,41,67)(14,68,42)(15,43,69)(16,70,44)(17,95,56)(18,57,96)(19,81,58)(20,59,82)(21,83,60)(22,61,84)(23,85,62)(24,63,86)(25,87,64)(26,49,88)(27,89,50)(28,51,90)(29,91,52)(30,53,92)(31,93,54)(32,55,94), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)>;

G:=Group( (1,25,45,87,71,64)(2,26,46,88,72,49)(3,27,47,89,73,50)(4,28,48,90,74,51)(5,29,33,91,75,52)(6,30,34,92,76,53)(7,31,35,93,77,54)(8,32,36,94,78,55)(9,17,37,95,79,56)(10,18,38,96,80,57)(11,19,39,81,65,58)(12,20,40,82,66,59)(13,21,41,83,67,60)(14,22,42,84,68,61)(15,23,43,85,69,62)(16,24,44,86,70,63), (1,45,71)(2,72,46)(3,47,73)(4,74,48)(5,33,75)(6,76,34)(7,35,77)(8,78,36)(9,37,79)(10,80,38)(11,39,65)(12,66,40)(13,41,67)(14,68,42)(15,43,69)(16,70,44)(17,95,56)(18,57,96)(19,81,58)(20,59,82)(21,83,60)(22,61,84)(23,85,62)(24,63,86)(25,87,64)(26,49,88)(27,89,50)(28,51,90)(29,91,52)(30,53,92)(31,93,54)(32,55,94), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96) );

G=PermutationGroup([(1,25,45,87,71,64),(2,26,46,88,72,49),(3,27,47,89,73,50),(4,28,48,90,74,51),(5,29,33,91,75,52),(6,30,34,92,76,53),(7,31,35,93,77,54),(8,32,36,94,78,55),(9,17,37,95,79,56),(10,18,38,96,80,57),(11,19,39,81,65,58),(12,20,40,82,66,59),(13,21,41,83,67,60),(14,22,42,84,68,61),(15,23,43,85,69,62),(16,24,44,86,70,63)], [(1,45,71),(2,72,46),(3,47,73),(4,74,48),(5,33,75),(6,76,34),(7,35,77),(8,78,36),(9,37,79),(10,80,38),(11,39,65),(12,66,40),(13,41,67),(14,68,42),(15,43,69),(16,70,44),(17,95,56),(18,57,96),(19,81,58),(20,59,82),(21,83,60),(22,61,84),(23,85,62),(24,63,86),(25,87,64),(26,49,88),(27,89,50),(28,51,90),(29,91,52),(30,53,92),(31,93,54),(32,55,94)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)])

144 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A···6F6G···6O8A···8H12A···12H12I···12T16A···16P24A···24P24Q···24AN48A···48AF
order12223333344446···66···68···812···1212···1216···1624···2424···2448···48
size11111122211111···12···21···11···12···23···31···12···23···3

144 irreducible representations

dim111111111111111122222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C8C8C12C12C16C24C24C48S3Dic3D6Dic3C3×S3C3⋊C8C3⋊C8C3×Dic3S3×C6C3×Dic3C3⋊C16C3×C3⋊C8C3×C3⋊C8C3×C3⋊C16
kernelC6×C3⋊C16C3×C3⋊C16C6×C24C2×C3⋊C16C3×C24C6×C12C3⋊C16C2×C24C3×C12C62C24C2×C12C3×C6C12C2×C6C6C2×C24C24C24C2×C12C2×C8C12C2×C6C8C8C2×C4C6C4C22C2
# reps121222424444168832111122222284416

Matrix representation of C6×C3⋊C16 in GL3(𝔽97) generated by

3600
0360
0036
,
100
0350
04361
,
9600
04760
0950
G:=sub<GL(3,GF(97))| [36,0,0,0,36,0,0,0,36],[1,0,0,0,35,43,0,0,61],[96,0,0,0,47,9,0,60,50] >;

C6×C3⋊C16 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes C_{16}
% in TeX

G:=Group("C6xC3:C16");
// GroupNames label

G:=SmallGroup(288,245);
// by ID

G=gap.SmallGroup(288,245);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,80,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽