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## G = C22×D5×A4order 480 = 25·3·5

### Direct product of C22, D5 and A4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×D5×A4
G = < a,b,c,d,e,f,g | a2=b2=c5=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, dcd=c-1, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 2356 in 356 conjugacy classes, 63 normal (15 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C23, C23, D5, D5, C10, C10, A4, C2×C6, C15, C24, C24, D10, D10, C2×C10, C2×C10, C2×A4, C2×A4, C22×C6, C3×D5, C30, C25, C22×D5, C22×D5, C22×D5, C22×C10, C22×C10, C22×A4, C22×A4, C5×A4, C6×D5, C2×C30, C23×D5, C23×D5, C23×C10, C23×A4, D5×A4, C10×A4, D5×C2×C6, D5×C24, C2×D5×A4, A4×C2×C10, C22×D5×A4
Quotients: C1, C2, C3, C22, C6, C23, D5, A4, C2×C6, D10, C2×A4, C22×C6, C3×D5, C22×D5, C22×A4, C6×D5, C23×A4, D5×A4, D5×C2×C6, C2×D5×A4, C22×D5×A4

Smallest permutation representation of C22×D5×A4
On 60 points
Generators in S60
(1 19)(2 20)(3 16)(4 17)(5 18)(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)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 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 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 51)(42 55)(43 54)(44 53)(45 52)(46 56)(47 60)(48 59)(49 58)(50 57)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 5A 5B 6A ··· 6F 6G ··· 6N 10A ··· 10F 10G ··· 10N 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 1 1 3 3 3 3 5 5 5 5 15 15 15 15 4 4 2 2 4 ··· 4 20 ··· 20 2 ··· 2 6 ··· 6 8 8 8 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 A4 C2×A4 C2×A4 D5×A4 C2×D5×A4 kernel C22×D5×A4 C2×D5×A4 A4×C2×C10 D5×C24 C23×D5 C23×C10 C22×A4 C2×A4 C24 C23 C22×D5 D10 C2×C10 C22 C2 # reps 1 6 1 2 12 2 2 6 4 12 1 6 1 2 6

Matrix representation of C22×D5×A4 in GL5(𝔽31)

 30 0 0 0 0 0 30 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 30 0 0 0 0 0 30 0 0 0 0 0 30
,
 12 1 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 12 0 0 0 0 30 0 0 0 0 0 30 0 0 0 0 0 30 0 0 0 0 0 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 30 0 0 0 0 0 30 0 0 0 25 5 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 30 0 0 0 0 0 1 0 0 0 0 26 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 6 26 29 0 0 0 0 5

G:=sub<GL(5,GF(31))| [30,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[12,30,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,12,30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,25,0,0,0,30,5,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,1,26,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,1,26,0,0,0,0,29,5] >;

C22×D5×A4 in GAP, Magma, Sage, TeX

C_2^2\times D_5\times A_4
% in TeX

G:=Group("C2^2xD5xA4");
// GroupNames label

G:=SmallGroup(480,1202);
// by ID

G=gap.SmallGroup(480,1202);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-5,340,152,18822]);
// 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,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations

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