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G = C24⋊C5order 80 = 24·5

The semidirect product of C24 and C5 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C24⋊C5, SmallGroup(80,49)

Series: Derived Chief Lower central Upper central

C1C24 — C24⋊C5
C1C24 — C24⋊C5
C24 — C24⋊C5
C1

Generators and relations for C24⋊C5
 G = < a,b,c,d,e | a2=b2=c2=d2=e5=1, ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, ede-1=a >

5C2
5C2
5C2
16C5
5C22
5C22
5C22
5C22
5C22
5C22
5C22
5C23
5C23
5C23

Character table of C24⋊C5

 class 12A2B2C5A5B5C5D
 size 155516161616
ρ111111111    trivial
ρ21111ζ52ζ5ζ54ζ53    linear of order 5
ρ31111ζ53ζ54ζ5ζ52    linear of order 5
ρ41111ζ54ζ52ζ53ζ5    linear of order 5
ρ51111ζ5ζ53ζ52ζ54    linear of order 5
ρ65-3110000    orthogonal faithful
ρ751-310000    orthogonal faithful
ρ8511-30000    orthogonal faithful

Permutation representations of C24⋊C5
On 10 points - transitive group 10T8
Generators in S10
(2 9)(3 10)(4 6)(5 7)
(4 6)(5 7)
(1 8)(4 6)
(1 8)(3 10)(4 6)(5 7)
(1 2 3 4 5)(6 7 8 9 10)

G:=sub<Sym(10)| (2,9)(3,10)(4,6)(5,7), (4,6)(5,7), (1,8)(4,6), (1,8)(3,10)(4,6)(5,7), (1,2,3,4,5)(6,7,8,9,10)>;

G:=Group( (2,9)(3,10)(4,6)(5,7), (4,6)(5,7), (1,8)(4,6), (1,8)(3,10)(4,6)(5,7), (1,2,3,4,5)(6,7,8,9,10) );

G=PermutationGroup([(2,9),(3,10),(4,6),(5,7)], [(4,6),(5,7)], [(1,8),(4,6)], [(1,8),(3,10),(4,6),(5,7)], [(1,2,3,4,5),(6,7,8,9,10)])

G:=TransitiveGroup(10,8);

On 16 points: primitive - transitive group 16T178
Generators in S16
(1 14)(2 15)(3 5)(4 9)(6 13)(7 11)(8 12)(10 16)
(1 5)(2 16)(3 14)(4 7)(6 8)(9 11)(10 15)(12 13)
(1 8)(2 11)(3 13)(4 10)(5 6)(7 15)(9 16)(12 14)
(1 15)(2 14)(3 16)(4 6)(5 10)(7 8)(9 13)(11 12)
(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)

G:=sub<Sym(16)| (1,14)(2,15)(3,5)(4,9)(6,13)(7,11)(8,12)(10,16), (1,5)(2,16)(3,14)(4,7)(6,8)(9,11)(10,15)(12,13), (1,8)(2,11)(3,13)(4,10)(5,6)(7,15)(9,16)(12,14), (1,15)(2,14)(3,16)(4,6)(5,10)(7,8)(9,13)(11,12), (2,3,4,5,6)(7,8,9,10,11)(12,13,14,15,16)>;

G:=Group( (1,14)(2,15)(3,5)(4,9)(6,13)(7,11)(8,12)(10,16), (1,5)(2,16)(3,14)(4,7)(6,8)(9,11)(10,15)(12,13), (1,8)(2,11)(3,13)(4,10)(5,6)(7,15)(9,16)(12,14), (1,15)(2,14)(3,16)(4,6)(5,10)(7,8)(9,13)(11,12), (2,3,4,5,6)(7,8,9,10,11)(12,13,14,15,16) );

G=PermutationGroup([(1,14),(2,15),(3,5),(4,9),(6,13),(7,11),(8,12),(10,16)], [(1,5),(2,16),(3,14),(4,7),(6,8),(9,11),(10,15),(12,13)], [(1,8),(2,11),(3,13),(4,10),(5,6),(7,15),(9,16),(12,14)], [(1,15),(2,14),(3,16),(4,6),(5,10),(7,8),(9,13),(11,12)], [(2,3,4,5,6),(7,8,9,10,11),(12,13,14,15,16)])

G:=TransitiveGroup(16,178);

On 20 points - transitive group 20T17
Generators in S20
(1 11)(2 12)(3 18)(4 14)(5 10)(6 16)(7 17)(8 13)(9 19)(15 20)
(1 6)(2 7)(3 18)(4 19)(8 13)(9 14)(11 16)(12 17)
(1 6)(3 13)(5 20)(8 18)(10 15)(11 16)
(1 6)(2 12)(3 13)(4 19)(5 15)(7 17)(8 18)(9 14)(10 20)(11 16)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,17);

On 20 points - transitive group 20T23
Generators in S20
(2 7)(3 13)(4 14)(5 10)(8 18)(9 19)(12 17)(15 20)
(1 16)(3 18)(4 9)(5 10)(6 11)(8 13)(14 19)(15 20)
(1 11)(2 17)(3 18)(4 14)(6 16)(7 12)(8 13)(9 19)
(1 6)(3 8)(4 14)(5 15)(9 19)(10 20)(11 16)(13 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,23);

C24⋊C5 is a maximal subgroup of   C24⋊D5  F16
C24⋊C5 is a maximal quotient of   2- 1+4⋊C5  C24⋊C25

Polynomial with Galois group C24⋊C5 over ℚ
actionf(x)Disc(f)
10T8x10-20x8+149x6-519x4+851x2-529210·118·236

Matrix representation of C24⋊C5 in GL5(ℤ)

10000
0-1000
00-100
000-10
0000-1
,
10000
0-1000
00-100
00010
00001
,
-10000
01000
00-100
00010
00001
,
-10000
0-1000
00-100
000-10
00001
,
01000
00100
00010
00001
10000

G:=sub<GL(5,Integers())| [1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C24⋊C5 in GAP, Magma, Sage, TeX

C_2^4\rtimes C_5
% in TeX

G:=Group("C2^4:C5");
// GroupNames label

G:=SmallGroup(80,49);
// by ID

G=gap.SmallGroup(80,49);
# by ID

G:=PCGroup([5,-5,-2,2,2,2,401,677,1103,1879]);
// Polycyclic

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

Export

Subgroup lattice of C24⋊C5 in TeX
Character table of C24⋊C5 in TeX

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