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## G = C3×M5(2)  order 96 = 25·3

### Direct product of C3 and M5(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×M5(2)
 Chief series C1 — C2 — C4 — C8 — C24 — C48 — C3×M5(2)
 Lower central C1 — C2 — C3×M5(2)
 Upper central C1 — C24 — C3×M5(2)

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

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 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E 12F 16A ··· 16H 24A ··· 24H 24I 24J 24K 24L 48A ··· 48P order 1 2 2 3 3 4 4 4 6 6 6 6 8 8 8 8 8 8 12 12 12 12 12 12 16 ··· 16 24 ··· 24 24 24 24 24 48 ··· 48 size 1 1 2 1 1 1 1 2 1 1 2 2 1 1 1 1 2 2 1 1 1 1 2 2 2 ··· 2 1 ··· 1 2 2 2 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C24 C24 M5(2) C3×M5(2) kernel C3×M5(2) C48 C2×C24 M5(2) C24 C2×C12 C16 C2×C8 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 4 4 8 8 4 8

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

 35 0 0 0 1 0 0 0 1
,
 1 0 0 0 33 95 0 36 64
,
 96 0 0 0 1 0 0 33 96
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

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