Copied to
clipboard

G = C3×2- 1+4order 96 = 25·3

Direct product of C3 and 2- 1+4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×2- 1+4, C6.20C24, C12.52C23, Q8(C3×D4), D4(C3×Q8), C4○D46C6, (C2×Q8)⋊7C6, (C6×Q8)⋊12C2, D4.4(C2×C6), Q8.7(C2×C6), (C2×C6).8C23, C2.5(C23×C6), C4.10(C22×C6), (C2×C12).71C22, (C3×D4).14C22, C22.3(C22×C6), (C3×Q8).15C22, (C3×D4)(C3×Q8), (C3×C4○D4)⋊9C2, (C2×C4).12(C2×C6), SmallGroup(96,225)

Series: Derived Chief Lower central Upper central

C1C2 — C3×2- 1+4
C1C2C6C2×C6C3×D4C3×C4○D4 — C3×2- 1+4
C1C2 — C3×2- 1+4
C1C6 — C3×2- 1+4

Generators and relations for C3×2- 1+4
 G = < a,b,c,d,e | a3=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2 [×5], C3, C4 [×10], C22 [×5], C6, C6 [×5], C2×C4 [×15], D4 [×10], Q8 [×10], C12 [×10], C2×C6 [×5], C2×Q8 [×5], C4○D4 [×10], C2×C12 [×15], C3×D4 [×10], C3×Q8 [×10], 2- 1+4, C6×Q8 [×5], C3×C4○D4 [×10], C3×2- 1+4
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C2×C6 [×35], C24, C22×C6 [×15], 2- 1+4, C23×C6, C3×2- 1+4

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

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

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

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

C3×2- 1+4 is a maximal subgroup of   2- 1+44S3  2- 1+4.2S3  D12.34C23  D12.35C23  D12.39C23  2- 1+4⋊C9
C3×2- 1+4 is a maximal quotient of   C3×D4×Q8

51 conjugacy classes

class 1 2A2B···2F3A3B4A···4J6A6B6C···6L12A···12T
order122···2334···4666···612···12
size112···2112···2112···22···2

51 irreducible representations

dim11111144
type+++-
imageC1C2C2C3C6C62- 1+4C3×2- 1+4
kernelC3×2- 1+4C6×Q8C3×C4○D42- 1+4C2×Q8C4○D4C3C1
# reps15102102012

Matrix representation of C3×2- 1+4 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
5626
5214
0046
0033
,
5060
1541
3020
6412
,
5650
5255
0031
0044
,
1614
1326
2546
1516
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,5,0,0,6,2,0,0,2,1,4,3,6,4,6,3],[5,1,3,6,0,5,0,4,6,4,2,1,0,1,0,2],[5,5,0,0,6,2,0,0,5,5,3,4,0,5,1,4],[1,1,2,1,6,3,5,5,1,2,4,1,4,6,6,6] >;

C3×2- 1+4 in GAP, Magma, Sage, TeX

C_3\times 2_-^{1+4}
% in TeX

G:=Group("C3xES-(2,2)");
// GroupNames label

G:=SmallGroup(96,225);
// by ID

G=gap.SmallGroup(96,225);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-2,601,295,476,230,1347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations

׿
×
𝔽