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G = C52⋊C4order 100 = 22·52

4th semidirect product of C52 and C4 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C52F5, C524C4, C5⋊D5.2C2, SmallGroup(100,12)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C4
C1C5C52C5⋊D5 — C52⋊C4
C52 — C52⋊C4
C1

Generators and relations for C52⋊C4
 G = < a,b,c | a5=b5=c4=1, ab=ba, cac-1=a2, cbc-1=b3 >

25C2
2C5
2C5
25C4
5D5
5D5
10D5
10D5
5F5
5F5

Character table of C52⋊C4

 class 124A4B5A5B5C5D5E5F
 size 1252525444444
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111111    linear of order 4
ρ41-1i-i111111    linear of order 4
ρ54000-1-1-14-1-1    orthogonal lifted from F5
ρ64000-1-1-1-14-1    orthogonal lifted from F5
ρ74000-1-5-1+53+5/2-1-13-5/2    orthogonal faithful
ρ840003+5/23-5/2-1+5-1-1-1-5    orthogonal faithful
ρ94000-1+5-1-53-5/2-1-13+5/2    orthogonal faithful
ρ1040003-5/23+5/2-1-5-1-1-1+5    orthogonal faithful

Permutation representations of C52⋊C4
On 10 points - transitive group 10T10
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 3 5 2 4)(6 9 7 10 8)
(1 6)(2 9 5 8)(3 7 4 10)

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

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

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

G:=TransitiveGroup(10,10);

On 20 points - transitive group 20T27
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 18 9 12)(2 16 8 14)(3 19 7 11)(4 17 6 13)(5 20 10 15)

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

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

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

G:=TransitiveGroup(20,27);

On 25 points - transitive group 25T10
Generators in S25
(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)
(1 25 19 14 9)(2 21 20 15 10)(3 22 16 11 6)(4 23 17 12 7)(5 24 18 13 8)
(2 4 5 3)(6 15 23 18)(7 13 22 20)(8 11 21 17)(9 14 25 19)(10 12 24 16)

G:=sub<Sym(25)| (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), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (2,4,5,3)(6,15,23,18)(7,13,22,20)(8,11,21,17)(9,14,25,19)(10,12,24,16)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (2,4,5,3)(6,15,23,18)(7,13,22,20)(8,11,21,17)(9,14,25,19)(10,12,24,16) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)], [(1,25,19,14,9),(2,21,20,15,10),(3,22,16,11,6),(4,23,17,12,7),(5,24,18,13,8)], [(2,4,5,3),(6,15,23,18),(7,13,22,20),(8,11,21,17),(9,14,25,19),(10,12,24,16)])

G:=TransitiveGroup(25,10);

C52⋊C4 is a maximal subgroup of
D5⋊F5  D5≀C2  C52⋊Q8  C52⋊Dic3  C152F5  He5⋊C4  C252F5  C536C4  C539C4
C52⋊C4 is a maximal quotient of
C525C8  C152F5  C252F5  He54C4  C536C4  C539C4

Polynomial with Galois group C52⋊C4 over ℚ
actionf(x)Disc(f)
10T10x10-5x9-40x8+165x7+560x6-1611x5-2485x4+4870x3-620x2-1080x+144224·326·515·112·194·712

Matrix representation of C52⋊C4 in GL4(𝔽41) generated by

354000
364000
104034
343477
,
40100
53500
614034
393577
,
00401
713934
00400
61400
G:=sub<GL(4,GF(41))| [35,36,1,34,40,40,0,34,0,0,40,7,0,0,34,7],[40,5,6,39,1,35,1,35,0,0,40,7,0,0,34,7],[0,7,0,6,0,1,0,1,40,39,40,40,1,34,0,0] >;

C52⋊C4 in GAP, Magma, Sage, TeX

C_5^2\rtimes C_4
% in TeX

G:=Group("C5^2:C4");
// GroupNames label

G:=SmallGroup(100,12);
// by ID

G=gap.SmallGroup(100,12);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,8,146,102,643,647]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C52⋊C4 in TeX
Character table of C52⋊C4 in TeX

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