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## G = C4×D5×A4order 480 = 25·3·5

### Direct product of C4, D5 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4×D5×A4
 Chief series C1 — C5 — C2×C10 — C22×C10 — C10×A4 — C2×D5×A4 — C4×D5×A4
 Lower central C2×C10 — C4×D5×A4
 Upper central C1 — C4

Generators and relations for C4×D5×A4
G = < a,b,c,d,e,f | a4=b5=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 780 in 132 conjugacy classes, 33 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, C23, C23, D5, D5, C10, C10, C12, A4, C2×C6, C15, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C12, C2×A4, C2×A4, C3×D5, C30, C23×C4, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C4×A4, C4×A4, C22×A4, C3×Dic5, C60, C5×A4, C6×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C2×C4×A4, D5×C12, D5×A4, C10×A4, D5×C22×C4, A4×Dic5, A4×C20, C2×D5×A4, C4×D5×A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D5, C12, A4, C2×C6, D10, C2×C12, C2×A4, C3×D5, C4×D5, C4×A4, C22×A4, C6×D5, C2×C4×A4, D5×C12, D5×A4, C2×D5×A4, C4×D5×A4

Smallest permutation representation of C4×D5×A4
On 60 points
Generators in S60
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)
(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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(1 9)(2 10)(3 6)(4 7)(5 8)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 10 10 10 10 10 10 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 3 3 5 5 15 15 4 4 1 1 3 3 5 5 15 15 2 2 4 4 20 20 20 20 2 2 6 6 6 6 4 4 4 4 20 20 20 20 8 8 8 8 2 2 2 2 6 6 6 6 8 8 8 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 6 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D5 D10 C3×D5 C4×D5 C6×D5 D5×C12 A4 C2×A4 C2×A4 C2×A4 C4×A4 D5×A4 C2×D5×A4 C4×D5×A4 kernel C4×D5×A4 A4×Dic5 A4×C20 C2×D5×A4 D5×C22×C4 D5×A4 C22×Dic5 C22×C20 C23×D5 C22×D5 C4×A4 C2×A4 C22×C4 A4 C23 C22 C4×D5 Dic5 C20 D10 D5 C4 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 2 2 4 4 4 8 1 1 1 1 4 2 2 4

Matrix representation of C4×D5×A4 in GL5(𝔽61)

 60 0 0 0 0 0 60 0 0 0 0 0 50 0 0 0 0 0 50 0 0 0 0 0 50
,
 18 1 0 0 0 42 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 60 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 36 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 45 1 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 13 6 0 0 0 0 48 1 0 0 0 14 0

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,50],[18,42,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,60,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,36,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,45,0,0,0,0,1,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,6,48,14,0,0,0,1,0] >;

C4×D5×A4 in GAP, Magma, Sage, TeX

C_4\times D_5\times A_4
% in TeX

G:=Group("C4xD5xA4");
// GroupNames label

G:=SmallGroup(480,1036);
// by ID

G=gap.SmallGroup(480,1036);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,92,648,271,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^5=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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