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## G = C3×M4(2)  order 48 = 24·3

### Direct product of C3 and M4(2)

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

Aliases: C3×M4(2), C83C6, C4.C12, C247C2, C12.4C4, C22.C12, C12.22C22, (C2×C6).1C4, (C2×C4).2C6, C4.6(C2×C6), C6.12(C2×C4), (C2×C12).8C2, C2.3(C2×C12), SmallGroup(48,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×M4(2)
 Chief series C1 — C2 — C4 — C12 — C24 — C3×M4(2)
 Lower central C1 — C2 — C3×M4(2)
 Upper central C1 — C12 — C3×M4(2)

Generators and relations for C3×M4(2)
G = < a,b,c | a3=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Character table of C3×M4(2)

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 1 1 1 1 2 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 ζ3 ζ32 1 1 -1 ζ3 ζ32 ζ6 ζ65 -1 1 1 -1 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 ζ3 ζ3 ζ32 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ6 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ7 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ8 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 -1 -1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ9 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 -1 -1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ10 1 1 -1 ζ3 ζ32 1 1 -1 ζ3 ζ32 ζ6 ζ65 1 -1 -1 1 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ11 1 1 -1 ζ32 ζ3 1 1 -1 ζ32 ζ3 ζ65 ζ6 1 -1 -1 1 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ12 1 1 -1 ζ32 ζ3 1 1 -1 ζ32 ζ3 ζ65 ζ6 -1 1 1 -1 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 ζ32 ζ32 ζ3 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ13 1 1 1 1 1 -1 -1 -1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 -1 -i i -i i -i i -i i linear of order 4 ρ14 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 i i -i -i -1 -1 -1 -1 1 1 i -i i -i -i i -i i linear of order 4 ρ15 1 1 1 1 1 -1 -1 -1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 -1 i -i i -i i -i i -i linear of order 4 ρ16 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -i -i i i -1 -1 -1 -1 1 1 -i i -i i i -i i -i linear of order 4 ρ17 1 1 -1 ζ32 ζ3 -1 -1 1 ζ32 ζ3 ζ65 ζ6 -i -i i i ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ18 1 1 1 ζ32 ζ3 -1 -1 -1 ζ32 ζ3 ζ3 ζ32 -i i -i i ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ19 1 1 -1 ζ3 ζ32 -1 -1 1 ζ3 ζ32 ζ6 ζ65 -i -i i i ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ20 1 1 -1 ζ32 ζ3 -1 -1 1 ζ32 ζ3 ζ65 ζ6 i i -i -i ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ21 1 1 1 ζ32 ζ3 -1 -1 -1 ζ32 ζ3 ζ3 ζ32 i -i i -i ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ22 1 1 1 ζ3 ζ32 -1 -1 -1 ζ3 ζ32 ζ32 ζ3 i -i i -i ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ23 1 1 -1 ζ3 ζ32 -1 -1 1 ζ3 ζ32 ζ6 ζ65 i i -i -i ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ24 1 1 1 ζ3 ζ32 -1 -1 -1 ζ3 ζ32 ζ32 ζ3 -i i -i i ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ25 2 -2 0 2 2 -2i 2i 0 -2 -2 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ26 2 -2 0 2 2 2i -2i 0 -2 -2 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ27 2 -2 0 -1-√-3 -1+√-3 2i -2i 0 1+√-3 1-√-3 0 0 0 0 0 0 2ζ4ζ3 2ζ43ζ3 2ζ4ζ32 2ζ43ζ32 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 2 -2 0 -1+√-3 -1-√-3 -2i 2i 0 1-√-3 1+√-3 0 0 0 0 0 0 2ζ43ζ32 2ζ4ζ32 2ζ43ζ3 2ζ4ζ3 0 0 0 0 0 0 0 0 0 0 complex faithful ρ29 2 -2 0 -1+√-3 -1-√-3 2i -2i 0 1-√-3 1+√-3 0 0 0 0 0 0 2ζ4ζ32 2ζ43ζ32 2ζ4ζ3 2ζ43ζ3 0 0 0 0 0 0 0 0 0 0 complex faithful ρ30 2 -2 0 -1-√-3 -1+√-3 -2i 2i 0 1+√-3 1-√-3 0 0 0 0 0 0 2ζ43ζ3 2ζ4ζ3 2ζ43ζ32 2ζ4ζ32 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×M4(2)
On 24 points - transitive group 24T16
Generators in S24
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 17)(8 9 18)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)

G:=sub<Sym(24)| (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)>;

G:=Group( (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24) );

G=PermutationGroup([[(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,17),(8,9,18)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24)]])

G:=TransitiveGroup(24,16);

C3×M4(2) is a maximal subgroup of
C12.53D4  C12.46D4  C12.47D4  D12⋊C4  D12.C4  C8⋊D6  C8.D6  M4(2).A4  C8⋊F7  C28.C12  He31M4(2)  He34M4(2)
C3×M4(2) is a maximal quotient of
C8⋊F7  C28.C12

Matrix representation of C3×M4(2) in GL2(𝔽13) generated by

 3 0 0 3
,
 0 6 3 0
,
 12 0 0 1
G:=sub<GL(2,GF(13))| [3,0,0,3],[0,3,6,0],[12,0,0,1] >;

C3×M4(2) in GAP, Magma, Sage, TeX

C_3\times M_4(2)
% in TeX

G:=Group("C3xM4(2)");
// GroupNames label

G:=SmallGroup(48,24);
// by ID

G=gap.SmallGroup(48,24);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,60,261,58]);
// Polycyclic

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

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