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G = C5×C32⋊C4  order 180 = 22·32·5

Direct product of C5 and C32⋊C4

direct product, metabelian, soluble, monomial, A-group

Aliases: C5×C32⋊C4, C32⋊C20, C3⋊S3.C10, (C3×C15)⋊5C4, (C5×C3⋊S3).1C2, SmallGroup(180,23)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C5×C32⋊C4
Chief series
C1 — C32 — C3⋊S3 — C5×C3⋊S3 — C5×C32⋊C4
Lower central
C32 — C5×C32⋊C4
Upper central
C1 — C5

Generators and relations for C5×C32⋊C4
 G = < a,b,c,d | a5=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

C1
9C2
2C3
2C3
C5
9C4
6S3
6S3
C32
9C10
2C15
2C15
C3⋊S3
9C20
6C5×S3
6C5×S3
C3×C15
C32⋊C4
C5×C3⋊S3
C5×C32⋊C4

Character table of C5×C32⋊C4

 class 123A3B4A4B5A5B5C5D10A10B10C10D15A15B15C15D15E15F15G15H20A20B20C20D20E20F20G20H
 size 194499111199994444444499999999
ρ1111111111111111111111111111111    trivial
ρ21111-1-11111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-111i-i1111-1-1-1-111111111ii-ii-ii-i-i    linear of order 4
ρ41-111-ii1111-1-1-1-111111111-i-ii-ii-iii    linear of order 4
ρ5111111ζ52ζ54ζ5ζ53ζ52ζ5ζ53ζ54ζ52ζ53ζ53ζ5ζ5ζ54ζ54ζ52ζ5ζ54ζ53ζ53ζ52ζ52ζ5ζ54    linear of order 5
ρ6111111ζ53ζ5ζ54ζ52ζ53ζ54ζ52ζ5ζ53ζ52ζ52ζ54ζ54ζ5ζ5ζ53ζ54ζ5ζ52ζ52ζ53ζ53ζ54ζ5    linear of order 5
ρ7111111ζ54ζ53ζ52ζ5ζ54ζ52ζ5ζ53ζ54ζ5ζ5ζ52ζ52ζ53ζ53ζ54ζ52ζ53ζ5ζ5ζ54ζ54ζ52ζ53    linear of order 5
ρ81111-1-1ζ53ζ5ζ54ζ52ζ53ζ54ζ52ζ5ζ53ζ52ζ52ζ54ζ54ζ5ζ5ζ53-ζ54-ζ5-ζ52-ζ52-ζ53-ζ53-ζ54-ζ5    linear of order 10
ρ91111-1-1ζ54ζ53ζ52ζ5ζ54ζ52ζ5ζ53ζ54ζ5ζ5ζ52ζ52ζ53ζ53ζ54-ζ52-ζ53-ζ5-ζ5-ζ54-ζ54-ζ52-ζ53    linear of order 10
ρ101111-1-1ζ52ζ54ζ5ζ53ζ52ζ5ζ53ζ54ζ52ζ53ζ53ζ5ζ5ζ54ζ54ζ52-ζ5-ζ54-ζ53-ζ53-ζ52-ζ52-ζ5-ζ54    linear of order 10
ρ111111-1-1ζ5ζ52ζ53ζ54ζ5ζ53ζ54ζ52ζ5ζ54ζ54ζ53ζ53ζ52ζ52ζ5-ζ53-ζ52-ζ54-ζ54-ζ5-ζ5-ζ53-ζ52    linear of order 10
ρ12111111ζ5ζ52ζ53ζ54ζ5ζ53ζ54ζ52ζ5ζ54ζ54ζ53ζ53ζ52ζ52ζ5ζ53ζ52ζ54ζ54ζ5ζ5ζ53ζ52    linear of order 5
ρ131-111-iiζ53ζ5ζ54ζ52-ζ53-ζ54-ζ52-ζ5ζ53ζ52ζ52ζ54ζ54ζ5ζ5ζ53ζ43ζ54ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ4ζ5    linear of order 20
ρ141-111-iiζ52ζ54ζ5ζ53-ζ52-ζ5-ζ53-ζ54ζ52ζ53ζ53ζ5ζ5ζ54ζ54ζ52ζ43ζ5ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ4ζ54    linear of order 20
ρ151-111i-iζ53ζ5ζ54ζ52-ζ53-ζ54-ζ52-ζ5ζ53ζ52ζ52ζ54ζ54ζ5ζ5ζ53ζ4ζ54ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ43ζ5    linear of order 20
ρ161-111i-iζ52ζ54ζ5ζ53-ζ52-ζ5-ζ53-ζ54ζ52ζ53ζ53ζ5ζ5ζ54ζ54ζ52ζ4ζ5ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ43ζ54    linear of order 20
ρ171-111i-iζ5ζ52ζ53ζ54-ζ5-ζ53-ζ54-ζ52ζ5ζ54ζ54ζ53ζ53ζ52ζ52ζ5ζ4ζ53ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ43ζ52    linear of order 20
ρ181-111-iiζ54ζ53ζ52ζ5-ζ54-ζ52-ζ5-ζ53ζ54ζ5ζ5ζ52ζ52ζ53ζ53ζ54ζ43ζ52ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ4ζ53    linear of order 20
ρ191-111i-iζ54ζ53ζ52ζ5-ζ54-ζ52-ζ5-ζ53ζ54ζ5ζ5ζ52ζ52ζ53ζ53ζ54ζ4ζ52ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ43ζ53    linear of order 20
ρ201-111-iiζ5ζ52ζ53ζ54-ζ5-ζ53-ζ54-ζ52ζ5ζ54ζ54ζ53ζ53ζ52ζ52ζ5ζ43ζ53ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ4ζ52    linear of order 20
ρ21401-200444400001-21-21-21-200000000    orthogonal lifted from C32⋊C4
ρ2240-210044440000-21-21-21-2100000000    orthogonal lifted from C32⋊C4
ρ2340-21004ζ544ζ534ζ524ζ50000-2ζ54ζ5-2ζ5ζ52-2ζ52ζ53-2ζ53ζ5400000000    complex faithful
ρ2440-21004ζ524ζ544ζ54ζ530000-2ζ52ζ53-2ζ53ζ5-2ζ5ζ54-2ζ54ζ5200000000    complex faithful
ρ25401-2004ζ534ζ54ζ544ζ520000ζ53-2ζ52ζ52-2ζ54ζ54-2ζ5ζ5-2ζ5300000000    complex faithful
ρ26401-2004ζ544ζ534ζ524ζ50000ζ54-2ζ5ζ5-2ζ52ζ52-2ζ53ζ53-2ζ5400000000    complex faithful
ρ27401-2004ζ524ζ544ζ54ζ530000ζ52-2ζ53ζ53-2ζ5ζ5-2ζ54ζ54-2ζ5200000000    complex faithful
ρ2840-21004ζ54ζ524ζ534ζ540000-2ζ5ζ54-2ζ54ζ53-2ζ53ζ52-2ζ52ζ500000000    complex faithful
ρ2940-21004ζ534ζ54ζ544ζ520000-2ζ53ζ52-2ζ52ζ54-2ζ54ζ5-2ζ5ζ5300000000    complex faithful
ρ30401-2004ζ54ζ524ζ534ζ540000ζ5-2ζ54ζ54-2ζ53ζ53-2ζ52ζ52-2ζ500000000    complex faithful

Permutation representations of C5×C32⋊C4
►On 30 points - transitive group 30T49
Generators in S30
(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)

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

(6 15 29)(7 11 30)(8 12 26)(9 13 27)(10 14 28)

(1 8)(2 9)(3 10)(4 6)(5 7)(11 22 30 16)(12 23 26 17)(13 24 27 18)(14 25 28 19)(15 21 29 20)
 
G:=sub<Sym(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), (1,17,23)(2,18,24)(3,19,25)(4,20,21)(5,16,22)(6,29,15)(7,30,11)(8,26,12)(9,27,13)(10,28,14), (6,15,29)(7,11,30)(8,12,26)(9,13,27)(10,14,28), (1,8)(2,9)(3,10)(4,6)(5,7)(11,22,30,16)(12,23,26,17)(13,24,27,18)(14,25,28,19)(15,21,29,20)>;
 
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), (1,17,23)(2,18,24)(3,19,25)(4,20,21)(5,16,22)(6,29,15)(7,30,11)(8,26,12)(9,27,13)(10,28,14), (6,15,29)(7,11,30)(8,12,26)(9,13,27)(10,14,28), (1,8)(2,9)(3,10)(4,6)(5,7)(11,22,30,16)(12,23,26,17)(13,24,27,18)(14,25,28,19)(15,21,29,20) );
 
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)], [(1,17,23),(2,18,24),(3,19,25),(4,20,21),(5,16,22),(6,29,15),(7,30,11),(8,26,12),(9,27,13),(10,28,14)], [(6,15,29),(7,11,30),(8,12,26),(9,13,27),(10,14,28)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,22,30,16),(12,23,26,17),(13,24,27,18),(14,25,28,19),(15,21,29,20)]])
 
G:=TransitiveGroup(30,49);
 

C5×C32⋊C4 is a maximal subgroup of   C5⋊2F9  C32⋊D20  C32⋊Dic10

Matrix representation of C5×C32⋊C4 ►in GL4(𝔽61) generated by

9000
0900
0090
0009
,
1000
0100
00601
00600
,
06000
16000
00601
00600
,
0010
0001
0100
1000
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,60,60,0,0,1,0],[0,1,0,0,60,60,0,0,0,0,60,60,0,0,1,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
 

C5×C32⋊C4 in GAP, Magma, Sage, TeX 

C_5\times C_3^2\rtimes C_4
 
% in TeX
 
G:=Group("C5xC3^2:C4");
 
// GroupNames label
 
G:=SmallGroup(180,23);
 
// by ID
 
G=gap.SmallGroup(180,23);
 
# by ID
 
G:=PCGroup([5,-2,-5,-2,-3,3,50,2803,93,4004,314]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
 
// generators/relations
 

Export

Subgroup lattice of C5×C32⋊C4 in TeX
Character table of C5×C32⋊C4 in TeX

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