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## G = (C22×D5)⋊A4order 480 = 25·3·5

### The semidirect product of C22×D5 and A4 acting faithfully

Aliases: (C22×D5)⋊A4, C242D5⋊C3, C5⋊(C24⋊C6), C22⋊A42D5, C243(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, C242D5, 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 1 2A 2B 2C 2D 3A 3B 4 5A 5B 6A 6B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 15A 15B 15C 15D size 1 3 6 6 20 16 16 60 2 2 80 80 6 6 6 6 6 6 6 6 6 6 32 32 32 32 ρ1 1 1 1 1 1 1 1 1 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 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 1 1 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ5 1 1 1 1 -1 ζ3 ζ32 -1 1 1 ζ6 ζ65 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ6 1 1 1 1 -1 ζ32 ζ3 -1 1 1 ζ65 ζ6 1 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ7 2 2 2 2 0 2 2 0 -1+√5/2 -1-√5/2 0 0 -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 ρ8 2 2 2 2 0 2 2 0 -1-√5/2 -1+√5/2 0 0 -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 ρ9 2 2 2 2 0 -1-√-3 -1+√-3 0 -1+√5/2 -1-√5/2 0 0 -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 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 complex lifted from C3×D5 ρ10 2 2 2 2 0 -1-√-3 -1+√-3 0 -1-√5/2 -1+√5/2 0 0 -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 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 complex lifted from C3×D5 ρ11 2 2 2 2 0 -1+√-3 -1-√-3 0 -1-√5/2 -1+√5/2 0 0 -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 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 complex lifted from C3×D5 ρ12 2 2 2 2 0 -1+√-3 -1-√-3 0 -1+√5/2 -1-√5/2 0 0 -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 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 complex lifted from C3×D5 ρ13 3 3 -1 -1 -3 0 0 1 3 3 0 0 -1 -1 -1 -1 3 -1 -1 -1 -1 3 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 -1 -1 3 0 0 -1 3 3 0 0 -1 -1 -1 -1 3 -1 -1 -1 -1 3 0 0 0 0 orthogonal lifted from A4 ρ15 6 -2 -2 2 0 0 0 0 6 6 0 0 -2 -2 -2 -2 -2 2 2 2 2 -2 0 0 0 0 orthogonal lifted from C24⋊C6 ρ16 6 -2 2 -2 0 0 0 0 6 6 0 0 2 2 2 2 -2 -2 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from C24⋊C6 ρ17 6 6 -2 -2 0 0 0 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -3-3√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -3+3√5/2 0 0 0 0 orthogonal lifted from D5×A4 ρ18 6 6 -2 -2 0 0 0 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -3+3√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -3-3√5/2 0 0 0 0 orthogonal lifted from D5×A4 ρ19 6 -2 -2 2 0 0 0 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -ζ53+3ζ52 3ζ54-ζ5 -ζ54+3ζ5 3ζ53-ζ52 1-√5/2 0 0 0 0 complex faithful ρ20 6 -2 2 -2 0 0 0 0 -3-3√5/2 -3+3√5/2 0 0 -ζ53+3ζ52 3ζ54-ζ5 -ζ54+3ζ5 3ζ53-ζ52 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 0 0 0 0 complex faithful ρ21 6 -2 -2 2 0 0 0 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -ζ54+3ζ5 -ζ53+3ζ52 3ζ53-ζ52 3ζ54-ζ5 1+√5/2 0 0 0 0 complex faithful ρ22 6 -2 2 -2 0 0 0 0 -3+3√5/2 -3-3√5/2 0 0 -ζ54+3ζ5 -ζ53+3ζ52 3ζ53-ζ52 3ζ54-ζ5 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 0 0 0 0 complex faithful ρ23 6 -2 2 -2 0 0 0 0 -3+3√5/2 -3-3√5/2 0 0 3ζ54-ζ5 3ζ53-ζ52 -ζ53+3ζ52 -ζ54+3ζ5 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 0 0 0 0 complex faithful ρ24 6 -2 -2 2 0 0 0 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 3ζ54-ζ5 3ζ53-ζ52 -ζ53+3ζ52 -ζ54+3ζ5 1+√5/2 0 0 0 0 complex faithful ρ25 6 -2 2 -2 0 0 0 0 -3-3√5/2 -3+3√5/2 0 0 3ζ53-ζ52 -ζ54+3ζ5 3ζ54-ζ5 -ζ53+3ζ52 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 0 0 0 0 complex faithful ρ26 6 -2 -2 2 0 0 0 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 3ζ53-ζ52 -ζ54+3ζ5 3ζ54-ζ5 -ζ53+3ζ52 1-√5/2 0 0 0 0 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)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 60 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 60 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 60
,
 40 58 0 0 0 0 0 0 1 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 41 0 0 0 0 0 0 39 54 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 60 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 60 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 60 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60 60 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60 60 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 60
,
 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60 60

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

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