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G = C2×C3⋊F5order 120 = 23·3·5

Direct product of C2 and C3⋊F5

Aliases: C2×C3⋊F5, C6⋊F5, C301C4, D5⋊Dic3, D10.S3, C10⋊Dic3, D5.2D6, C5⋊(C2×Dic3), C32(C2×F5), C152(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 >

Character table of C2×C3⋊F5

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6A 6B 6C 10 15A 15B 30A 30B size 1 1 5 5 2 15 15 15 15 4 2 10 10 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 i i -i -i 1 1 -1 -1 1 1 1 1 1 linear of order 4 ρ6 1 -1 -1 1 1 -i i i -i 1 -1 1 -1 -1 1 1 -1 -1 linear of order 4 ρ7 1 -1 -1 1 1 i -i -i i 1 -1 1 -1 -1 1 1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 1 -i -i i i 1 1 -1 -1 1 1 1 1 1 linear of order 4 ρ9 2 -2 2 -2 -1 0 0 0 0 2 1 1 -1 -2 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 0 0 0 0 2 -1 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 -2 -2 2 -1 0 0 0 0 2 1 -1 1 -2 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 2 -2 -2 -1 0 0 0 0 2 -1 1 1 2 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ13 4 -4 0 0 4 0 0 0 0 -1 -4 0 0 1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ14 4 4 0 0 4 0 0 0 0 -1 4 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ15 4 -4 0 0 -2 0 0 0 0 -1 2 0 0 1 1-√-15/2 1+√-15/2 -1-√-15/2 -1+√-15/2 complex faithful ρ16 4 4 0 0 -2 0 0 0 0 -1 -2 0 0 -1 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ17 4 4 0 0 -2 0 0 0 0 -1 -2 0 0 -1 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ18 4 -4 0 0 -2 0 0 0 0 -1 2 0 0 1 1+√-15/2 1-√-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  C158M4(2)  D10.D6

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

 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 56 0 0 0 0 36 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 60 60 60 60
,
 45 56 0 0 0 0 51 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 60 60 60 60

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

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