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## G = C5×D4×A4order 480 = 25·3·5

### Direct product of C5, D4 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C5×D4×A4
 Chief series C1 — C22 — C23 — C22×C10 — C10×A4 — A4×C2×C10 — C5×D4×A4
 Lower central C22 — C23 — C5×D4×A4
 Upper central C1 — C10 — C5×D4

Generators and relations for C5×D4×A4
G = < a,b,c,d,e,f | a5=b4=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: 416 in 132 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C2×C4, D4, D4, C23, C23, C10, C10, C12, A4, C2×C6, C15, C22×C4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C3×D4, C2×A4, C2×A4, C30, C22×D4, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, C4×A4, C22×A4, C60, C5×A4, C2×C30, C22×C20, D4×C10, C23×C10, D4×A4, D4×C15, C10×A4, C10×A4, D4×C2×C10, A4×C20, A4×C2×C10, C5×D4×A4
Quotients: C1, C2, C3, C22, C5, C6, D4, C10, A4, C2×C6, C15, C2×C10, C3×D4, C2×A4, C30, C5×D4, C22×A4, C5×A4, C2×C30, D4×A4, D4×C15, C10×A4, A4×C2×C10, C5×D4×A4

Smallest permutation representation of C5×D4×A4
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 43 13 37)(2 44 14 38)(3 45 15 39)(4 41 11 40)(5 42 12 36)(6 31 60 30)(7 32 56 26)(8 33 57 27)(9 34 58 28)(10 35 59 29)(16 52 22 46)(17 53 23 47)(18 54 24 48)(19 55 25 49)(20 51 21 50)
(1 37)(2 38)(3 39)(4 40)(5 36)(6 31)(7 32)(8 33)(9 34)(10 35)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)(36 42)(37 43)(38 44)(39 45)(40 41)(46 52)(47 53)(48 54)(49 55)(50 51)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 60)(7 56)(8 57)(9 58)(10 59)(26 32)(27 33)(28 34)(29 35)(30 31)(36 42)(37 43)(38 44)(39 45)(40 41)
(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 26 16)(6 51 41)(7 52 42)(8 53 43)(9 54 44)(10 55 45)(11 31 21)(12 32 22)(13 33 23)(14 34 24)(15 35 25)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 10E ··· 10L 10M ··· 10T 10U ··· 10AB 12A 12B 15A ··· 15H 20A 20B 20C 20D 20E 20F 20G 20H 30A ··· 30H 30I ··· 30X 60A ··· 60H order 1 2 2 2 2 2 2 2 3 3 4 4 5 5 5 5 6 6 6 6 6 6 10 10 10 10 10 ··· 10 10 ··· 10 10 ··· 10 12 12 15 ··· 15 20 20 20 20 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 2 3 3 6 6 4 4 2 6 1 1 1 1 4 4 8 8 8 8 1 1 1 1 2 ··· 2 3 ··· 3 6 ··· 6 8 8 4 ··· 4 2 2 2 2 6 6 6 6 4 ··· 4 8 ··· 8 8 ··· 8

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 6 6 type + + + + + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 D4 C3×D4 C5×D4 D4×C15 A4 C2×A4 C2×A4 C5×A4 C10×A4 C10×A4 D4×A4 C5×D4×A4 kernel C5×D4×A4 A4×C20 A4×C2×C10 D4×C2×C10 D4×A4 C22×C20 C23×C10 C4×A4 C22×A4 C22×D4 C22×C4 C24 C5×A4 C2×C10 A4 C22 C5×D4 C20 C2×C10 D4 C4 C22 C5 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 2 4 8 1 1 2 4 4 8 1 4

Matrix representation of C5×D4×A4 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 34 0 0 0 0 0 34 0 0 0 0 0 34
,
 0 60 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 0 60 0 0 0 60 0 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 0 60 1 0 0 0 60 0 0 0 1 60 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 60 0 0 1 0 60 0 0 0 0 60
,
 47 0 0 0 0 0 47 0 0 0 0 0 0 0 47 0 0 47 0 0 0 0 0 47 0

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34],[0,1,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[0,60,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[47,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,47,0,0,47,0,0] >;

C5×D4×A4 in GAP, Magma, Sage, TeX

C_5\times D_4\times A_4
% in TeX

G:=Group("C5xD4xA4");
// GroupNames label

G:=SmallGroup(480,1127);
// by ID

G=gap.SmallGroup(480,1127);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,2,869,2539,4430]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^4=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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