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G = C3×C8⋊D6order 288 = 25·32

Direct product of C3 and C8⋊D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C8⋊D6, D242C6, C2411D6, C12.89D12, C62.63D4, C81(S3×C6), C241(C2×C6), C24⋊C21C6, C4○D122C6, (C3×D24)⋊3C2, (C6×D12)⋊9C2, D124(C2×C6), (C2×D12)⋊7C6, C6.13(C6×D4), (C3×C24)⋊3C22, Dic64(C2×C6), (C2×C6).47D12, C2.15(C6×D12), C4.14(C3×D12), (C3×C12).82D4, C12.12(C3×D4), (C2×C12).238D6, C6.101(C2×D12), M4(2)⋊1(C3×S3), (C3×M4(2))⋊3S3, (C3×M4(2))⋊1C6, C22.5(C3×D12), (C3×D12)⋊39C22, C3216(C8⋊C22), C12.32(C22×C6), (C6×C12).115C22, C12.219(C22×S3), (C3×C12).164C23, (C3×Dic6)⋊37C22, (C32×M4(2))⋊1C2, C4.30(S3×C2×C6), C31(C3×C8⋊C22), (C2×C6).6(C3×D4), (C3×C24⋊C2)⋊3C2, (C3×C4○D12)⋊6C2, (C2×C4).13(S3×C6), (C2×C12).26(C2×C6), (C3×C6).183(C2×D4), SmallGroup(288,679)

Series: Derived Chief Lower central Upper central

C1C12 — C3×C8⋊D6
C1C3C6C12C3×C12C3×D12C6×D12 — C3×C8⋊D6
C3C6C12 — C3×C8⋊D6
C1C6C2×C12C3×M4(2)

Generators and relations for C3×C8⋊D6
 G = < a,b,c,d | a3=b8=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 442 in 146 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×3], C6 [×2], C6 [×7], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×5], C2×C6 [×2], C2×C6 [×6], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3×C6, C3×C6, C24 [×4], C24 [×2], Dic6, C4×S3, D12, D12 [×2], D12, C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×5], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12 [×2], S3×C6 [×5], C62, C24⋊C2 [×2], D24 [×2], C3×M4(2) [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C2×D12, C4○D12, C6×D4, C3×C4○D4, C3×C24 [×2], C3×Dic6, S3×C12, C3×D12, C3×D12 [×2], C3×D12, C3×C3⋊D4, C6×C12, S3×C2×C6, C8⋊D6, C3×C8⋊C22, C3×C24⋊C2 [×2], C3×D24 [×2], C32×M4(2), C6×D12, C3×C4○D12, C3×C8⋊D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C8⋊C22, S3×C6 [×3], C2×D12, C6×D4, C3×D12 [×2], S3×C2×C6, C8⋊D6, C3×C8⋊C22, C6×D12, C3×C8⋊D6

Smallest permutation representation of C3×C8⋊D6
On 48 points
Generators in S48
(1 36 29)(2 37 30)(3 38 31)(4 39 32)(5 40 25)(6 33 26)(7 34 27)(8 35 28)(9 43 24)(10 44 17)(11 45 18)(12 46 19)(13 47 20)(14 48 21)(15 41 22)(16 42 23)
(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)
(1 29 36)(2 26 37 6 30 33)(3 31 38)(4 28 39 8 32 35)(5 25 40)(7 27 34)(9 47 24 13 43 20)(10 44 17)(11 41 18 15 45 22)(12 46 19)(14 48 21)(16 42 23)
(1 19)(2 18)(3 17)(4 24)(5 23)(6 22)(7 21)(8 20)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)

G:=sub<Sym(48)| (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,43,24)(10,44,17)(11,45,18)(12,46,19)(13,47,20)(14,48,21)(15,41,22)(16,42,23), (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), (1,29,36)(2,26,37,6,30,33)(3,31,38)(4,28,39,8,32,35)(5,25,40)(7,27,34)(9,47,24,13,43,20)(10,44,17)(11,41,18,15,45,22)(12,46,19)(14,48,21)(16,42,23), (1,19)(2,18)(3,17)(4,24)(5,23)(6,22)(7,21)(8,20)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)>;

G:=Group( (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,43,24)(10,44,17)(11,45,18)(12,46,19)(13,47,20)(14,48,21)(15,41,22)(16,42,23), (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), (1,29,36)(2,26,37,6,30,33)(3,31,38)(4,28,39,8,32,35)(5,25,40)(7,27,34)(9,47,24,13,43,20)(10,44,17)(11,41,18,15,45,22)(12,46,19)(14,48,21)(16,42,23), (1,19)(2,18)(3,17)(4,24)(5,23)(6,22)(7,21)(8,20)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43) );

G=PermutationGroup([(1,36,29),(2,37,30),(3,38,31),(4,39,32),(5,40,25),(6,33,26),(7,34,27),(8,35,28),(9,43,24),(10,44,17),(11,45,18),(12,46,19),(13,47,20),(14,48,21),(15,41,22),(16,42,23)], [(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)], [(1,29,36),(2,26,37,6,30,33),(3,31,38),(4,28,39,8,32,35),(5,25,40),(7,27,34),(9,47,24,13,43,20),(10,44,17),(11,41,18,15,45,22),(12,46,19),(14,48,21),(16,42,23)], [(1,19),(2,18),(3,17),(4,24),(5,23),(6,22),(7,21),(8,20),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,40),(25,42),(26,41),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)])

63 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C6A6B6C···6G6H6I6J6K···6P8A8B12A···12J12K12L12M12N12O24A···24P
order12222233333444666···66666···68812···12121212121224···24
size112121212112222212112···244412···12442···244412124···4

63 irreducible representations

dim111111111111222222222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C3×S3D12C3×D4D12C3×D4S3×C6S3×C6C3×D12C3×D12C8⋊C22C8⋊D6C3×C8⋊C22C3×C8⋊D6
kernelC3×C8⋊D6C3×C24⋊C2C3×D24C32×M4(2)C6×D12C3×C4○D12C8⋊D6C24⋊C2D24C3×M4(2)C2×D12C4○D12C3×M4(2)C3×C12C62C24C2×C12M4(2)C12C12C2×C6C2×C6C8C2×C4C4C22C32C3C3C1
# reps122111244222111212222242441224

Matrix representation of C3×C8⋊D6 in GL4(𝔽73) generated by

8000
0800
0080
0008
,
0100
27000
00027
00720
,
64000
0900
0080
00065
,
0080
00065
64000
0900
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,27,0,0,1,0,0,0,0,0,0,72,0,0,27,0],[64,0,0,0,0,9,0,0,0,0,8,0,0,0,0,65],[0,0,64,0,0,0,0,9,8,0,0,0,0,65,0,0] >;

C3×C8⋊D6 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_6
% in TeX

G:=Group("C3xC8:D6");
// GroupNames label

G:=SmallGroup(288,679);
// by ID

G=gap.SmallGroup(288,679);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,142,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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