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G = C2×C3⋊F5  order 120 = 23·3·5

Direct product of C2 and C3⋊F5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C3⋊F5, C6⋊F5, C30⋊1C4, D5⋊Dic3, D10.S3, C10⋊Dic3, D5.2D6, C5⋊(C2×Dic3), C3⋊2(C2×F5), C15⋊2(C2×C4), (C3×D5)⋊2C4, (C6×D5).2C2, (C3×D5).2C22, SmallGroup(120,41)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C2×C3⋊F5
Chief series
C1 — C5 — C15 — C3×D5 — C3⋊F5 — C2×C3⋊F5
Lower central
C15 — C2×C3⋊F5
Upper central
C1 — C2

Generators and relations for C2×C3⋊F5
 G = < a,b,c,d | a2=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

C1
C2
5C2
5C2
C3
C5
5C22
15C4
15C4
C6
5C6
5C6
D5
C10
D5
C15
15C2×C4
5Dic3
5Dic3
5C2×C6
D10
3F5
3F5
C3×D5
C3×D5
C30
5C2×Dic3
3C2×F5
C3⋊F5
C3⋊F5
C6×D5
C2×C3⋊F5

Character table of C2×C3⋊F5

 class 12A2B2C34A4B4C4D56A6B6C1015A15B30A30B
 size 115521515151542101044444
ρ1111111111111111111    trivial
ρ21-11-111-11-11-1-11-111-1-1    linear of order 2
ρ311111-1-1-1-1111111111    linear of order 2
ρ41-11-11-11-111-1-11-111-1-1    linear of order 2
ρ511-1-11ii-i-i11-1-111111    linear of order 4
ρ61-1-111-iii-i1-11-1-111-1-1    linear of order 4
ρ71-1-111i-i-ii1-11-1-111-1-1    linear of order 4
ρ811-1-11-i-iii11-1-111111    linear of order 4
ρ92-22-2-10000211-1-2-1-111    orthogonal lifted from D6
ρ102222-100002-1-1-12-1-1-1-1    orthogonal lifted from S3
ρ112-2-22-1000021-11-2-1-111    symplectic lifted from Dic3, Schur index 2
ρ1222-2-2-100002-1112-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ134-40040000-1-4001-1-111    orthogonal lifted from C2×F5
ρ14440040000-1400-1-1-1-1-1    orthogonal lifted from F5
ρ154-400-20000-120011-√-15/21+√-15/2-1-√-15/2-1+√-15/2    complex faithful
ρ164400-20000-1-200-11-√-15/21+√-15/21+√-15/21-√-15/2    complex lifted from C3⋊F5
ρ174400-20000-1-200-11+√-15/21-√-15/21-√-15/21+√-15/2    complex lifted from C3⋊F5
ρ184-400-20000-120011+√-15/21-√-15/2-1+√-15/2-1-√-15/2    complex faithful

Permutation representations of C2×C3⋊F5
►On 30 points - transitive group 30T17
Generators in S30
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)

(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)

(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)(26 27 28 29 30)

(2 3 5 4)(6 13 7 15)(8 12 10 11)(9 14)(16 18 17 20)(21 28 22 30)(23 27 25 26)(24 29)
 
G:=sub<Sym(30)| (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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)(26,27,28,29,30), (2,3,5,4)(6,13,7,15)(8,12,10,11)(9,14)(16,18,17,20)(21,28,22,30)(23,27,25,26)(24,29)>;
 
G:=Group( (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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)(26,27,28,29,30), (2,3,5,4)(6,13,7,15)(8,12,10,11)(9,14)(16,18,17,20)(21,28,22,30)(23,27,25,26)(24,29) );
 
G=PermutationGroup([[(1,19),(2,20),(3,16),(4,17),(5,18),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(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),(26,27,28,29,30)], [(2,3,5,4),(6,13,7,15),(8,12,10,11),(9,14),(16,18,17,20),(21,28,22,30),(23,27,25,26),(24,29)]])
 
G:=TransitiveGroup(30,17);
 

C2×C3⋊F5 is a maximal subgroup of   Dic3×F5  D6⋊F5  Dic3⋊F5  C60⋊C4  D10.D6  C2×S3×F5  D10.S4
C2×C3⋊F5 is a maximal quotient of   C60.C4  C12.F5  C60⋊C4  C15⋊8M4(2)  D10.D6

Matrix representation of C2×C3⋊F5 ►in GL6(𝔽61)

6000000
0600000
001000
000100
000010
000001
,
45560000
36150000
001000
000100
000010
000001
,
100000
010000
000100
000010
000001
0060606060
,
45560000
51160000
001000
000001
000100
0060606060

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[45,36,0,0,0,0,56,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[45,51,0,0,0,0,56,16,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;
 

C2×C3⋊F5 in GAP, Magma, Sage, TeX 

C_2\times C_3\rtimes F_5
 
% in TeX
 
G:=Group("C2xC3:F5");
 
// GroupNames label
 
G:=SmallGroup(120,41);
 
// by ID
 
G=gap.SmallGroup(120,41);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-3,-5,20,323,1804,614]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C3⋊F5 in TeX
Character table of C2×C3⋊F5 in TeX

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