Copied to
clipboard

G = C3×C12.46D4order 288 = 25·32

Direct product of C3 and C12.46D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12.46D4, C12.86D12, (C6×D12).5C2, (C2×D12).6C6, (C3×C12).40D4, C12.55(C3×D4), C4.11(C3×D12), (C22×S3).C12, C4.Dic32C6, C6.49(D6⋊C4), (C2×C12).220D6, (C3×M4(2))⋊7S3, M4(2)⋊3(C3×S3), (C3×M4(2))⋊7C6, C62.37(C2×C4), C22.4(S3×C12), (C6×C12).44C22, C325(C4.D4), C12.138(C3⋊D4), (C32×M4(2))⋊11C2, (S3×C2×C6).2C4, (C2×C4).1(S3×C6), C2.9(C3×D6⋊C4), (C2×C6).59(C4×S3), (C2×C6).2(C2×C12), C31(C3×C4.D4), C4.21(C3×C3⋊D4), C6.8(C3×C22⋊C4), (C2×C12).14(C2×C6), (C3×C4.Dic3)⋊18C2, (C3×C6).48(C22⋊C4), SmallGroup(288,257)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C12.46D4
C1C3C6C2×C6C2×C12C6×C12C6×D12 — C3×C12.46D4
C3C6C2×C6 — C3×C12.46D4
C1C6C2×C12C3×M4(2)

Generators and relations for C3×C12.46D4
 G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >

Subgroups: 314 in 102 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×6], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×5], M4(2), M4(2), C2×D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8, C24 [×5], D12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4.D4, C3×C12 [×2], S3×C6 [×4], C62, C4.Dic3, C3×M4(2) [×2], C3×M4(2) [×2], C2×D12, C6×D4, C3×C3⋊C8, C3×C24, C3×D12 [×2], C6×C12, S3×C2×C6 [×2], C12.46D4, C3×C4.D4, C3×C4.Dic3, C32×M4(2), C6×D12, C3×C12.46D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C4.D4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C12.46D4, C3×C4.D4, C3×D6⋊C4, C3×C12.46D4

Smallest permutation representation of C3×C12.46D4
On 48 points
Generators in S48
(1 27 39)(2 28 40)(3 29 33)(4 30 34)(5 31 35)(6 32 36)(7 25 37)(8 26 38)(9 46 22)(10 47 23)(11 48 24)(12 41 17)(13 42 18)(14 43 19)(15 44 20)(16 45 21)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)
(1 39 27)(2 40 28)(3 33 29)(4 34 30)(5 35 31)(6 36 32)(7 37 25)(8 38 26)(9 46 22)(10 47 23)(11 48 24)(12 41 17)(13 42 18)(14 43 19)(15 44 20)(16 45 21)
(1 13)(2 10)(3 11)(4 16)(5 9)(6 14)(7 15)(8 12)(17 38)(18 39)(19 36)(20 37)(21 34)(22 35)(23 40)(24 33)(25 44)(26 41)(27 42)(28 47)(29 48)(30 45)(31 46)(32 43)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B6A6B6C···6G6H6I6J6K6L6M6N8A8B8C8D12A···12J12K12L12M24A···24P24Q24R24S24T
order122223333344666···66666666888812···1212121224···2424242424
size11212121122222112···2444121212124412122···24444···412121212

63 irreducible representations

dim11111111112222222222224444
type++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D4D6C3×S3D12C3⋊D4C3×D4C4×S3S3×C6C3×D12C3×C3⋊D4S3×C12C4.D4C12.46D4C3×C4.D4C3×C12.46D4
kernelC3×C12.46D4C3×C4.Dic3C32×M4(2)C6×D12C12.46D4S3×C2×C6C4.Dic3C3×M4(2)C2×D12C22×S3C3×M4(2)C3×C12C2×C12M4(2)C12C12C12C2×C6C2×C4C4C4C22C32C3C3C1
# reps11112422281212224224441224

Matrix representation of C3×C12.46D4 in GL4(𝔽73) generated by

8000
0800
0080
0008
,
0100
27000
0001
00460
,
1000
07200
0010
00072
,
64000
06400
0080
0008
,
0010
0001
1000
0100
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,27,0,0,1,0,0,0,0,0,0,46,0,0,1,0],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C3×C12.46D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{46}D_4
% in TeX

G:=Group("C3xC12.46D4");
// GroupNames label

G:=SmallGroup(288,257);
// by ID

G=gap.SmallGroup(288,257);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,136,1271,9414]);
// Polycyclic

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

׿
×
𝔽