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## G = C4×S3×F5order 480 = 25·3·5

### Direct product of C4, S3 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C4×S3×F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — C2×S3×F5 — C4×S3×F5
 Lower central C15 — C4×S3×F5
 Upper central C1 — C4

Generators and relations for C4×S3×F5
G = < a,b,c,d,e | a4=b3=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 1012 in 216 conjugacy classes, 80 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22×C4, Dic5, Dic5, C20, C20, F5, F5, D10, D10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×C42, C4×D5, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C4×Dic3, C4×C12, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C4×F5, C4×F5, C2×C4×D5, C22×F5, S3×C42, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, S3×F5, C6×F5, C2×C3⋊F5, C2×S3×D5, C2×C4×F5, Dic3×F5, C12×F5, C4×C3⋊F5, C4×S3×D5, C2×S3×F5, C4×S3×F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, F5, C4×S3, C22×S3, C2×C42, C2×F5, S3×C2×C4, C4×F5, C22×F5, S3×C42, S3×F5, C2×C4×F5, C2×S3×F5, C4×S3×F5

Smallest permutation representation of C4×S3×F5
On 60 points
Generators in S60
(1 46 16 31)(2 47 17 32)(3 48 18 33)(4 49 19 34)(5 50 20 35)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)(31 46)(32 47)(33 48)(34 49)(35 50)(36 56)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 53)(44 54)(45 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 16)(2 18 5 19)(3 20 4 17)(6 21)(7 23 10 24)(8 25 9 22)(11 26)(12 28 15 29)(13 30 14 27)(31 46)(32 48 35 49)(33 50 34 47)(36 51)(37 53 40 54)(38 55 39 52)(41 56)(42 58 45 59)(43 60 44 57)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 8 8 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 S3 D6 D6 C4×S3 C4×S3 C4×S3 F5 C2×F5 C2×F5 C2×F5 C4×F5 S3×F5 C2×S3×F5 C4×S3×F5 kernel C4×S3×F5 Dic3×F5 C12×F5 C4×C3⋊F5 C4×S3×D5 C2×S3×F5 S3×Dic5 D30.C2 S3×C20 C4×D15 S3×F5 C4×F5 C4×D5 C2×F5 Dic5 C20 F5 C4×S3 Dic3 C12 D6 S3 C4 C2 C1 # reps 1 2 1 1 1 2 2 2 2 2 16 1 1 2 2 2 8 1 1 1 1 4 1 1 2

Matrix representation of C4×S3×F5 in GL6(𝔽61)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 60 60 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 60 60 60 60

G:=sub<GL(6,GF(61))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;

C4×S3×F5 in GAP, Magma, Sage, TeX

C_4\times S_3\times F_5
% in TeX

G:=Group("C4xS3xF5");
// GroupNames label

G:=SmallGroup(480,994);
// by ID

G=gap.SmallGroup(480,994);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,100,1356,9414,2379]);
// Polycyclic

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

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