Copied to
clipboard

G = C3×M5(2)  order 96 = 25·3

Direct product of C3 and M5(2)

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

Aliases: C3×M5(2), C4.C24, C487C2, C163C6, C12.4C8, C24.6C4, C8.2C12, C22.C24, C24.29C22, C8.8(C2×C6), (C2×C8).8C6, (C2×C6).1C8, C6.13(C2×C8), (C2×C4).5C12, C2.3(C2×C24), C12.49(C2×C4), (C2×C12).14C4, (C2×C24).18C2, C4.12(C2×C12), SmallGroup(96,60)

Series: Derived Chief Lower central Upper central

C1C2 — C3×M5(2)
C1C2C4C8C24C48 — C3×M5(2)
C1C2 — C3×M5(2)
C1C24 — C3×M5(2)

Generators and relations for C3×M5(2)
 G = < a,b,c | a3=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

2C2
2C6

Smallest permutation representation of C3×M5(2)
On 48 points
Generators in S48
(1 20 48)(2 21 33)(3 22 34)(4 23 35)(5 24 36)(6 25 37)(7 26 38)(8 27 39)(9 28 40)(10 29 41)(11 30 42)(12 31 43)(13 32 44)(14 17 45)(15 18 46)(16 19 47)
(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 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)

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

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

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

C3×M5(2) is a maximal subgroup of
C24.97D4  C48⋊C4  C24.Q8  C8.25D12  Dic6.C8  M5(2)⋊S3  C12.4D8  D242C4  C16.12D6  C16⋊D6  C16.D6

60 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D8A8B8C8D8E8F12A12B12C12D12E12F16A···16H24A···24H24I24J24K24L48A···48P
order12233444666688888812121212121216···1624···242424242448···48
size1121111211221111221111222···21···122222···2

60 irreducible representations

dim1111111111111122
type+++
imageC1C2C2C3C4C4C6C6C8C8C12C12C24C24M5(2)C3×M5(2)
kernelC3×M5(2)C48C2×C24M5(2)C24C2×C12C16C2×C8C12C2×C6C8C2×C4C4C22C3C1
# reps1212224244448848

Matrix representation of C3×M5(2) in GL3(𝔽97) generated by

3500
010
001
,
100
03395
03664
,
9600
010
03396
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[1,0,0,0,33,36,0,95,64],[96,0,0,0,1,33,0,0,96] >;

C3×M5(2) in GAP, Magma, Sage, TeX

C_3\times M_5(2)
% in TeX

G:=Group("C3xM5(2)");
// GroupNames label

G:=SmallGroup(96,60);
// by ID

G=gap.SmallGroup(96,60);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,72,601,69,88]);
// Polycyclic

G:=Group<a,b,c|a^3=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

Export

Subgroup lattice of C3×M5(2) in TeX

׿
×
𝔽