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G = C60⋊C4order 240 = 24·3·5

1st semidirect product of C60 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C601C4, C121F5, D5.1D12, C201Dic3, D10.13D6, D5.2Dic6, Dic53Dic3, C4⋊(C3⋊F5), C5⋊(C4⋊Dic3), C31(C4⋊F5), C152(C4⋊C4), (C4×D5).4S3, (C3×D5).3D4, C6.11(C2×F5), (C3×D5).2Q8, C30.11(C2×C4), (C3×Dic5)⋊5C4, (D5×C12).6C2, C10.4(C2×Dic3), (C6×D5).20C22, C2.5(C2×C3⋊F5), (C2×C3⋊F5).3C2, SmallGroup(240,121)

Series: Derived Chief Lower central Upper central

C1C30 — C60⋊C4
C1C5C15C3×D5C6×D5C2×C3⋊F5 — C60⋊C4
C15C30 — C60⋊C4
C1C2C4

Generators and relations for C60⋊C4
 G = < a,b | a60=b4=1, bab-1=a47 >

5C2
5C2
5C4
5C22
30C4
30C4
5C6
5C6
5C2×C4
15C2×C4
15C2×C4
5C12
5C2×C6
10Dic3
10Dic3
6F5
6F5
15C4⋊C4
5C2×Dic3
5C2×Dic3
5C2×C12
3C2×F5
3C2×F5
2C3⋊F5
2C3⋊F5
5C4⋊Dic3
3C4⋊F5

Character table of C60⋊C4

 class 12A2B2C34A4B4C4D4E4F56A6B6C1012A12B12C12D15A15B20A20B30A30B60A60B60C60D
 size 115522103030303042101042210104444444444
ρ1111111111111111111111111111111    trivial
ρ211111-1-11-11-111111-1-1-1-111-1-111-1-1-1-1    linear of order 2
ρ311111-1-1-11-1111111-1-1-1-111-1-111-1-1-1-1    linear of order 2
ρ41111111-1-1-1-11111111111111111111    linear of order 2
ρ511-1-11-11-iii-i11-1-11-1-11111-1-111-1-1-1-1    linear of order 4
ρ611-1-111-1-i-iii11-1-1111-1-11111111111    linear of order 4
ρ711-1-111-1ii-i-i11-1-1111-1-11111111111    linear of order 4
ρ811-1-11-11i-i-ii11-1-11-1-11111-1-111-1-1-1-1    linear of order 4
ρ92222-1-2-200002-1-1-121111-1-1-2-2-1-11111    orthogonal lifted from D6
ρ102-22-220000002-2-22-200002200-2-20000    orthogonal lifted from D4
ρ112222-12200002-1-1-12-1-1-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ122-22-2-1000000211-1-23-33-3-1-100113-33-3    orthogonal lifted from D12
ρ132-22-2-1000000211-1-2-33-33-1-10011-33-33    orthogonal lifted from D12
ρ1422-2-2-12-200002-1112-1-111-1-122-1-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ1522-2-2-1-2200002-111211-1-1-1-1-2-2-1-11111    symplectic lifted from Dic3, Schur index 2
ρ162-2-22-100000021-11-23-3-33-1-100113-33-3    symplectic lifted from Dic6, Schur index 2
ρ172-2-2220000002-22-2-200002200-2-20000    symplectic lifted from Q8, Schur index 2
ρ182-2-22-100000021-11-2-333-3-1-10011-33-33    symplectic lifted from Dic6, Schur index 2
ρ1944004-400000-1400-1-4-400-1-111-1-11111    orthogonal lifted from C2×F5
ρ2044004400000-1400-14400-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ214400-2400000-1-200-1-2-2001--15/21+-15/2-1-11+-15/21--15/21--15/21+-15/21+-15/21--15/2    complex lifted from C3⋊F5
ρ224400-2-400000-1-200-122001+-15/21--15/2111--15/21+-15/2-1--15/2-1+-15/2-1+-15/2-1--15/2    complex lifted from C2×C3⋊F5
ρ234400-2400000-1-200-1-2-2001+-15/21--15/2-1-11--15/21+-15/21+-15/21--15/21--15/21+-15/2    complex lifted from C3⋊F5
ρ244400-2-400000-1-200-122001--15/21+-15/2111+-15/21--15/2-1+-15/2-1--15/2-1--15/2-1+-15/2    complex lifted from C2×C3⋊F5
ρ254-4004000000-1-40010000-1-1-5--511-5-5--5--5    complex lifted from C4⋊F5
ρ264-4004000000-1-40010000-1-1--5-511--5--5-5-5    complex lifted from C4⋊F5
ρ274-400-2000000-12001-2323001+-15/21--15/2-5--5-1+-15/2-1--15/24ζ34ζ534ζ5243ζ343ζ5443ζ543ζ3243ζ5343ζ524ζ324ζ544ζ5    complex faithful
ρ284-400-2000000-12001-2323001--15/21+-15/2--5-5-1--15/2-1+-15/243ζ3243ζ5343ζ524ζ324ζ544ζ54ζ34ζ534ζ5243ζ343ζ5443ζ5    complex faithful
ρ294-400-2000000-1200123-23001+-15/21--15/2--5-5-1+-15/2-1--15/24ζ324ζ544ζ543ζ3243ζ5343ζ5243ζ343ζ5443ζ54ζ34ζ534ζ52    complex faithful
ρ304-400-2000000-1200123-23001--15/21+-15/2-5--5-1--15/2-1+-15/243ζ343ζ5443ζ54ζ34ζ534ζ524ζ324ζ544ζ543ζ3243ζ5343ζ52    complex faithful

Smallest permutation representation of C60⋊C4
On 60 points
Generators in S60
(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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)
(1 16)(2 39 50 3)(4 25 28 37)(5 48 17 24)(6 11)(7 34 55 58)(8 57 44 45)(9 20 33 32)(10 43 22 19)(12 29 60 53)(13 52 49 40)(14 15 38 27)(18 47 54 35)(21 56)(23 42 59 30)(26 51)(31 46)(36 41)

G:=sub<Sym(60)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,16)(2,39,50,3)(4,25,28,37)(5,48,17,24)(6,11)(7,34,55,58)(8,57,44,45)(9,20,33,32)(10,43,22,19)(12,29,60,53)(13,52,49,40)(14,15,38,27)(18,47,54,35)(21,56)(23,42,59,30)(26,51)(31,46)(36,41)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,16)(2,39,50,3)(4,25,28,37)(5,48,17,24)(6,11)(7,34,55,58)(8,57,44,45)(9,20,33,32)(10,43,22,19)(12,29,60,53)(13,52,49,40)(14,15,38,27)(18,47,54,35)(21,56)(23,42,59,30)(26,51)(31,46)(36,41) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,16),(2,39,50,3),(4,25,28,37),(5,48,17,24),(6,11),(7,34,55,58),(8,57,44,45),(9,20,33,32),(10,43,22,19),(12,29,60,53),(13,52,49,40),(14,15,38,27),(18,47,54,35),(21,56),(23,42,59,30),(26,51),(31,46),(36,41)])

C60⋊C4 is a maximal subgroup of
D12⋊F5  Dic30⋊C4  Dic5.Dic6  Dic5.4Dic6  C120⋊C4  D5.D24  D20⋊Dic3  Dic102Dic3  F5×Dic6  C4⋊F53S3  F5×D12  S3×C4⋊F5  (C2×C12)⋊6F5  D4×C3⋊F5  Q8×C3⋊F5
C60⋊C4 is a maximal quotient of
C120⋊C4  D5.D24  C40.Dic3  C24.1F5  C60⋊C8  Dic5.13D12  D10.10D12

Matrix representation of C60⋊C4 in GL6(𝔽61)

0600000
110000
000291645
003229450
001604529
0032451629
,
5000000
11110000
000163229
002903245
004532029
002932160

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,0,32,16,32,0,0,29,29,0,45,0,0,16,45,45,16,0,0,45,0,29,29],[50,11,0,0,0,0,0,11,0,0,0,0,0,0,0,29,45,29,0,0,16,0,32,32,0,0,32,32,0,16,0,0,29,45,29,0] >;

C60⋊C4 in GAP, Magma, Sage, TeX

C_{60}\rtimes C_4
% in TeX

G:=Group("C60:C4");
// GroupNames label

G:=SmallGroup(240,121);
// by ID

G=gap.SmallGroup(240,121);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,964,5189,1745]);
// Polycyclic

G:=Group<a,b|a^60=b^4=1,b*a*b^-1=a^47>;
// generators/relations

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Subgroup lattice of C60⋊C4 in TeX
Character table of C60⋊C4 in TeX

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