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G = C24⋊C2order 48 = 24·3

2nd semidirect product of C24 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82S3, C242C2, C6.1D4, C4.8D6, C31SD16, C2.3D12, Dic61C2, D12.1C2, C12.8C22, SmallGroup(48,6)

Series: Derived Chief Lower central Upper central

C1C12 — C24⋊C2
C1C3C6C12D12 — C24⋊C2
C3C6C12 — C24⋊C2
C1C2C4C8

Generators and relations for C24⋊C2
 G = < a,b | a24=b2=1, bab=a11 >

12C2
6C22
6C4
4S3
3Q8
3D4
2Dic3
2D6
3SD16

Character table of C24⋊C2

 class 12A2B34A4B68A8B12A12B24A24B24C24D
 size 11122212222222222
ρ1111111111111111    trivial
ρ211-111-1111111111    linear of order 2
ρ311111-11-1-111-1-1-1-1    linear of order 2
ρ411-11111-1-111-1-1-1-1    linear of order 2
ρ5220-120-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ6220-120-1-2-2-1-11111    orthogonal lifted from D6
ρ72202-20200-2-20000    orthogonal lifted from D4
ρ8220-1-20-10011-333-3    orthogonal lifted from D12
ρ9220-1-20-100113-3-33    orthogonal lifted from D12
ρ102-20200-2--2-200-2-2--2--2    complex lifted from SD16
ρ112-20200-2-2--200--2--2-2-2    complex lifted from SD16
ρ122-20-1001--2-2-33ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32    complex faithful
ρ132-20-1001-2--2-33ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ87ζ328785ζ32    complex faithful
ρ142-20-1001--2-23-3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ83ζ3838ζ3    complex faithful
ρ152-20-1001-2--23-3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3    complex faithful

Permutation representations of C24⋊C2
On 24 points - transitive group 24T35
Generators in S24
(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 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)

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

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

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

G:=TransitiveGroup(24,35);

Matrix representation of C24⋊C2 in GL2(𝔽11) generated by

01
15
,
75
84
G:=sub<GL(2,GF(11))| [0,1,1,5],[7,8,5,4] >;

C24⋊C2 in GAP, Magma, Sage, TeX

C_{24}\rtimes C_2
% in TeX

G:=Group("C24:C2");
// GroupNames label

G:=SmallGroup(48,6);
// by ID

G=gap.SmallGroup(48,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,61,26,182,42,804]);
// Polycyclic

G:=Group<a,b|a^24=b^2=1,b*a*b=a^11>;
// generators/relations

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