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G = (C22×D5)⋊A4  order 480 = 25·3·5

The semidirect product of C22×D5 and A4 acting faithfully

metabelian, soluble, monomial

Aliases: (C22×D5)⋊A4, C24⋊2D5⋊C3, C5⋊(C24⋊C6), C22⋊A4⋊2D5, C24⋊3(C3×D5), (C23×C10)⋊3C6, C22.4(D5×A4), (C5×C22⋊A4)⋊3C2, (C2×C10).4(C2×A4), SmallGroup(480,268)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23×C10 — (C22×D5)⋊A4
Chief series
C1 — C5 — C2×C10 — C23×C10 — C5×C22⋊A4 — (C22×D5)⋊A4
Lower central
C23×C10 — (C22×D5)⋊A4
Upper central
C1

Generators and relations for (C22×D5)⋊A4
 G = < a,b,c,d,e,f,g | a2=b2=c5=d2=e2=f2=g3=1, gbg-1=ab=ba, ac=ca, fdf=gdg-1=ad=da, ae=ea, af=fa, gag-1=b, bc=cb, ede=bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, cg=gc, geg-1=ef=fe, gfg-1=e >

Subgroups: 568 in 70 conjugacy classes, 11 normal (all characteristic)
C1, C2, C3, C4, C22, C22, C5, C6, C2×C4, D4, C23, D5, C10, A4, C15, C22⋊C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, C2×A4, C3×D5, C22≀C2, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22⋊A4, C5×A4, C23.D5, C2×C5⋊D4, C23×C10, C24⋊C6, D5×A4, C24⋊2D5, C5×C22⋊A4, (C22×D5)⋊A4
Quotients: C1, C2, C3, C6, D5, A4, C2×A4, C3×D5, C24⋊C6, D5×A4, (C22×D5)⋊A4

Character table of (C22×D5)⋊A4

 class 12A2B2C2D3A3B45A5B6A6B10A10B10C10D10E10F10G10H10I10J15A15B15C15D
 size 136620161660228080666666666632323232
ρ111111111111111111111111111    trivial
ρ21111-111-111-1-111111111111111    linear of order 2
ρ311111ζ3ζ32111ζ32ζ31111111111ζ3ζ3ζ32ζ32    linear of order 3
ρ411111ζ32ζ3111ζ3ζ321111111111ζ32ζ32ζ3ζ3    linear of order 3
ρ51111-1ζ3ζ32-111ζ6ζ651111111111ζ3ζ3ζ32ζ32    linear of order 6
ρ61111-1ζ32ζ3-111ζ65ζ61111111111ζ32ζ32ζ3ζ3    linear of order 6
ρ722220220-1+√5/2-1-√5/200-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ822220220-1-√5/2-1+√5/200-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ922220-1-√-3-1+√-30-1+√5/2-1-√5/200-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52    complex lifted from C3×D5
ρ1022220-1-√-3-1+√-30-1-√5/2-1+√5/200-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5    complex lifted from C3×D5
ρ1122220-1+√-3-1-√-30-1-√5/2-1+√5/200-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5    complex lifted from C3×D5
ρ1222220-1+√-3-1-√-30-1+√5/2-1-√5/200-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52    complex lifted from C3×D5
ρ1333-1-1-30013300-1-1-1-13-1-1-1-130000    orthogonal lifted from C2×A4
ρ1433-1-1300-13300-1-1-1-13-1-1-1-130000    orthogonal lifted from A4
ρ156-2-2200006600-2-2-2-2-22222-20000    orthogonal lifted from C24⋊C6
ρ166-22-2000066002222-2-2-2-2-2-20000    orthogonal lifted from C24⋊C6
ρ1766-2-20000-3-3√5/2-3+3√5/2001+√5/21-√5/21-√5/21+√5/2-3-3√5/21+√5/21-√5/21-√5/21+√5/2-3+3√5/20000    orthogonal lifted from D5×A4
ρ1866-2-20000-3+3√5/2-3-3√5/2001-√5/21+√5/21+√5/21-√5/2-3+3√5/21-√5/21+√5/21+√5/21-√5/2-3-3√5/20000    orthogonal lifted from D5×A4
ρ196-2-220000-3-3√5/2-3+3√5/2001+√5/21-√5/21-√5/21+√5/21+√5/2-ζ53+3ζ523ζ54-ζ5-ζ54+3ζ53ζ53-ζ521-√5/20000    complex faithful
ρ206-22-20000-3-3√5/2-3+3√5/200-ζ53+3ζ523ζ54-ζ5-ζ54+3ζ53ζ53-ζ521+√5/21+√5/21-√5/21-√5/21+√5/21-√5/20000    complex faithful
ρ216-2-220000-3+3√5/2-3-3√5/2001-√5/21+√5/21+√5/21-√5/21-√5/2-ζ54+3ζ5-ζ53+3ζ523ζ53-ζ523ζ54-ζ51+√5/20000    complex faithful
ρ226-22-20000-3+3√5/2-3-3√5/200-ζ54+3ζ5-ζ53+3ζ523ζ53-ζ523ζ54-ζ51-√5/21-√5/21+√5/21+√5/21-√5/21+√5/20000    complex faithful
ρ236-22-20000-3+3√5/2-3-3√5/2003ζ54-ζ53ζ53-ζ52-ζ53+3ζ52-ζ54+3ζ51-√5/21-√5/21+√5/21+√5/21-√5/21+√5/20000    complex faithful
ρ246-2-220000-3+3√5/2-3-3√5/2001-√5/21+√5/21+√5/21-√5/21-√5/23ζ54-ζ53ζ53-ζ52-ζ53+3ζ52-ζ54+3ζ51+√5/20000    complex faithful
ρ256-22-20000-3-3√5/2-3+3√5/2003ζ53-ζ52-ζ54+3ζ53ζ54-ζ5-ζ53+3ζ521+√5/21+√5/21-√5/21-√5/21+√5/21-√5/20000    complex faithful
ρ266-2-220000-3-3√5/2-3+3√5/2001+√5/21-√5/21-√5/21+√5/21+√5/23ζ53-ζ52-ζ54+3ζ53ζ54-ζ5-ζ53+3ζ521-√5/20000    complex faithful

Smallest permutation representation of (C22×D5)⋊A4
►On 40 points
Generators in S40
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)

(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

(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)

(1 28)(2 27)(3 26)(4 30)(5 29)(6 23)(7 22)(8 21)(9 25)(10 24)(11 38)(12 37)(13 36)(14 40)(15 39)(16 33)(17 32)(18 31)(19 35)(20 34)

(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

(6 16 11)(7 17 12)(8 18 13)(9 19 14)(10 20 15)(26 36 31)(27 37 32)(28 38 33)(29 39 34)(30 40 35)
 
G:=sub<Sym(40)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,23)(7,22)(8,21)(9,25)(10,24)(11,38)(12,37)(13,36)(14,40)(15,39)(16,33)(17,32)(18,31)(19,35)(20,34), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (6,16,11)(7,17,12)(8,18,13)(9,19,14)(10,20,15)(26,36,31)(27,37,32)(28,38,33)(29,39,34)(30,40,35)>;
 
G:=Group( (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,23)(7,22)(8,21)(9,25)(10,24)(11,38)(12,37)(13,36)(14,40)(15,39)(16,33)(17,32)(18,31)(19,35)(20,34), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (6,16,11)(7,17,12)(8,18,13)(9,19,14)(10,20,15)(26,36,31)(27,37,32)(28,38,33)(29,39,34)(30,40,35) );
 
G=PermutationGroup([[(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)], [(1,28),(2,27),(3,26),(4,30),(5,29),(6,23),(7,22),(8,21),(9,25),(10,24),(11,38),(12,37),(13,36),(14,40),(15,39),(16,33),(17,32),(18,31),(19,35),(20,34)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(6,16,11),(7,17,12),(8,18,13),(9,19,14),(10,20,15),(26,36,31),(27,37,32),(28,38,33),(29,39,34),(30,40,35)]])
 

Matrix representation of (C22×D5)⋊A4 ►in GL8(𝔽61)

10000000
01000000
00606060000
00001000
00010000
00000606060
00000001
00000010
,
10000000
01000000
00010000
00100000
00606060000
00000010
00000100
00000606060
,
4058000000
13000000
00100000
00010000
00001000
00000100
00000010
00000001
,
741000000
3954000000
00000010
00000100
00000606060
00010000
00100000
00606060000
,
10000000
01000000
00606060000
00001000
00010000
00000001
00000606060
00000100
,
10000000
01000000
00001000
00606060000
00100000
00000010
00000100
00000606060
,
130000000
013000000
00100000
00001000
00606060000
00000100
00000001
00000606060

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,60],[40,1,0,0,0,0,0,0,58,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[7,39,0,0,0,0,0,0,41,54,0,0,0,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,60,0,0,0,1,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,60,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,60],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60] >;
 

(C22×D5)⋊A4 in GAP, Magma, Sage, TeX 

(C_2^2\times D_5)\rtimes A_4
 
% in TeX
 
G:=Group("(C2^2xD5):A4");
 
// GroupNames label
 
G:=SmallGroup(480,268);
 
// by ID
 
G=gap.SmallGroup(480,268);
 
# by ID
 
G:=PCGroup([7,-2,-3,-2,2,-2,2,-5,1640,198,1683,94,851,1524,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^5=d^2=e^2=f^2=g^3=1,g*b*g^-1=a*b=b*a,a*c=c*a,f*d*f=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,g*a*g^-1=b,b*c=c*b,e*d*e=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,c*g=g*c,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
 
// generators/relations
 

Export

Character table of (C22×D5)⋊A4 in TeX

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