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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

C1C2 — C3×M4(2)
C1C2C4C12C24 — C3×M4(2)
C1C2 — C3×M4(2)
C1C12 — C3×M4(2)

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

2C2
2C6

Character table of C3×M4(2)

 class 12A2B3A3B4A4B4C6A6B6C6D8A8B8C8D12A12B12C12D12E12F24A24B24C24D24E24F24G24H
 size 112111121122222211112222222222
ρ1111111111111111111111111111111    trivial
ρ211-11111-111-1-11-1-111111-1-1-1-1-1-11111    linear of order 2
ρ3111111111111-1-1-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-11111-111-1-1-111-11111-1-11111-1-1-1-1    linear of order 2
ρ511-1ζ3ζ3211-1ζ3ζ32ζ6ζ65-111-1ζ32ζ32ζ3ζ3ζ6ζ65ζ3ζ3ζ32ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ6111ζ32ζ3111ζ32ζ3ζ3ζ321111ζ3ζ3ζ32ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ7111ζ3ζ32111ζ3ζ32ζ32ζ31111ζ32ζ32ζ3ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ8111ζ32ζ3111ζ32ζ3ζ3ζ32-1-1-1-1ζ3ζ3ζ32ζ32ζ3ζ32ζ6ζ6ζ65ζ65ζ65ζ65ζ6ζ6    linear of order 6
ρ9111ζ3ζ32111ζ3ζ32ζ32ζ3-1-1-1-1ζ32ζ32ζ3ζ3ζ32ζ3ζ65ζ65ζ6ζ6ζ6ζ6ζ65ζ65    linear of order 6
ρ1011-1ζ3ζ3211-1ζ3ζ32ζ6ζ651-1-11ζ32ζ32ζ3ζ3ζ6ζ65ζ65ζ65ζ6ζ6ζ32ζ32ζ3ζ3    linear of order 6
ρ1111-1ζ32ζ311-1ζ32ζ3ζ65ζ61-1-11ζ3ζ3ζ32ζ32ζ65ζ6ζ6ζ6ζ65ζ65ζ3ζ3ζ32ζ32    linear of order 6
ρ1211-1ζ32ζ311-1ζ32ζ3ζ65ζ6-111-1ζ3ζ3ζ32ζ32ζ65ζ6ζ32ζ32ζ3ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ1311111-1-1-11111i-ii-i-1-1-1-1-1-1-ii-ii-ii-ii    linear of order 4
ρ1411-111-1-1111-1-1ii-i-i-1-1-1-111i-ii-i-ii-ii    linear of order 4
ρ1511111-1-1-11111-ii-ii-1-1-1-1-1-1i-ii-ii-ii-i    linear of order 4
ρ1611-111-1-1111-1-1-i-iii-1-1-1-111-ii-iii-ii-i    linear of order 4
ρ1711-1ζ32ζ62-1-11ζ32ζ62ζ65ζ6ζ2ζ2ζ2ζ2ζ65ζ65ζ6ζ6ζ62ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ18111ζ32ζ62-1-1-1ζ32ζ62ζ62ζ32ζ2ζ2ζ2ζ2ζ65ζ65ζ6ζ6ζ65ζ6ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ1911-1ζ62ζ32-1-11ζ62ζ32ζ6ζ65ζ2ζ2ζ2ζ2ζ6ζ6ζ65ζ65ζ32ζ62ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ2011-1ζ32ζ62-1-11ζ32ζ62ζ65ζ6ζ2ζ2ζ2ζ2ζ65ζ65ζ6ζ6ζ62ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ21111ζ32ζ62-1-1-1ζ32ζ62ζ62ζ32ζ2ζ2ζ2ζ2ζ65ζ65ζ6ζ6ζ65ζ6ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ22111ζ62ζ32-1-1-1ζ62ζ32ζ32ζ62ζ2ζ2ζ2ζ2ζ6ζ6ζ65ζ65ζ6ζ65ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ2311-1ζ62ζ32-1-11ζ62ζ32ζ6ζ65ζ2ζ2ζ2ζ2ζ6ζ6ζ65ζ65ζ32ζ62ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ24111ζ62ζ32-1-1-1ζ62ζ32ζ32ζ62ζ2ζ2ζ2ζ2ζ6ζ6ζ65ζ65ζ6ζ65ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ252-2022-2i2i0-2-2000000-2i2i-2i2i0000000000    complex lifted from M4(2)
ρ262-20222i-2i0-2-20000002i-2i2i-2i0000000000    complex lifted from M4(2)
ρ272-20-1--3-1+-32i-2i01+-31--30000004ζ343ζ34ζ3243ζ320000000000    complex faithful
ρ282-20-1+-3-1--3-2i2i01--31+-300000043ζ324ζ3243ζ34ζ30000000000    complex faithful
ρ292-20-1+-3-1--32i-2i01--31+-30000004ζ3243ζ324ζ343ζ30000000000    complex faithful
ρ302-20-1--3-1+-3-2i2i01+-31--300000043ζ34ζ343ζ324ζ320000000000    complex faithful

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

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

30
03
,
06
30
,
120
01
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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