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## G = C22×S3×F5order 480 = 25·3·5

### Direct product of C22, S3 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C22×S3×F5
 Chief series C1 — C5 — C15 — C3×D5 — C3×F5 — S3×F5 — C2×S3×F5 — C22×S3×F5
 Lower central C15 — C22×S3×F5
 Upper central C1 — C22

Generators and relations for C22×S3×F5
G = < a,b,c,d,e,f | a2=b2=c3=d2=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 2292 in 472 conjugacy classes, 166 normal (20 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×8], C22, C22 [×34], C5, S3 [×4], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×28], C23 [×15], D5, D5 [×3], D5 [×4], C10 [×3], C10 [×4], Dic3 [×4], C12 [×4], D6 [×6], D6 [×22], C2×C6, C2×C6 [×6], C15, C22×C4 [×14], C24, F5 [×4], F5 [×4], D10 [×6], D10 [×22], C2×C10, C2×C10 [×6], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×S3 [×13], C22×C6, C5×S3 [×4], C3×D5, C3×D5 [×3], D15 [×4], C30 [×3], C23×C4, C2×F5 [×6], C2×F5 [×22], C22×D5, C22×D5 [×13], C22×C10, S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, C3×F5 [×4], C3⋊F5 [×4], S3×D5 [×16], C6×D5 [×6], S3×C10 [×6], D30 [×6], C2×C30, C22×F5, C22×F5 [×13], C23×D5, S3×C22×C4, S3×F5 [×16], C6×F5 [×6], C2×C3⋊F5 [×6], C2×S3×D5 [×12], D5×C2×C6, S3×C2×C10, C22×D15, C23×F5, C2×S3×F5 [×12], C2×C6×F5, C22×C3⋊F5, C22×S3×D5, C22×S3×F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, F5, C4×S3 [×4], C22×S3 [×7], C23×C4, C2×F5 [×7], S3×C2×C4 [×6], S3×C23, C22×F5 [×7], S3×C22×C4, S3×F5, C23×F5, C2×S3×F5 [×3], C22×S3×F5

Smallest permutation representation of C22×S3×F5
On 60 points
Generators in S60
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(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 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)
(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 46)(2 47)(3 48)(4 49)(5 50)(6 56)(7 57)(8 58)(9 59)(10 60)(11 51)(12 52)(13 53)(14 54)(15 55)(16 31)(17 32)(18 33)(19 34)(20 35)(21 41)(22 42)(23 43)(24 44)(25 45)(26 36)(27 37)(28 38)(29 39)(30 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)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)
(1 46)(2 48 5 49)(3 50 4 47)(6 51)(7 53 10 54)(8 55 9 52)(11 56)(12 58 15 59)(13 60 14 57)(16 31)(17 33 20 34)(18 35 19 32)(21 36)(22 38 25 39)(23 40 24 37)(26 41)(27 43 30 44)(28 45 29 42)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A ··· 4H 4I ··· 4P 5 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 10G 12A ··· 12H 15 30A 30B 30C order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 5 6 6 6 6 6 6 6 10 10 10 10 10 10 10 12 ··· 12 15 30 30 30 size 1 1 1 1 3 3 3 3 5 5 5 5 15 15 15 15 2 5 ··· 5 15 ··· 15 4 2 2 2 10 10 10 10 4 4 4 12 12 12 12 10 ··· 10 8 8 8 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 C4×S3 C4×S3 F5 C2×F5 C2×F5 S3×F5 C2×S3×F5 kernel C22×S3×F5 C2×S3×F5 C2×C6×F5 C22×C3⋊F5 C22×S3×D5 C2×S3×D5 S3×C2×C10 C22×D15 C22×F5 C2×F5 C22×D5 D10 C2×C10 C22×S3 D6 C2×C6 C22 C2 # reps 1 12 1 1 1 12 2 2 1 6 1 6 2 1 6 1 1 3

Matrix representation of C22×S3×F5 in GL8(𝔽61)

 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 0 60 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
,
 60 0 0 0 0 0 0 0 0 60 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
,
 0 60 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 1 0 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
,
 60 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 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
,
 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 0 0 0 60 0 0 0 0 1 0 0 60 0 0 0 0 0 1 0 60 0 0 0 0 0 0 1 60
,
 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C22×S3×F5 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times F_5
% in TeX

G:=Group("C2^2xS3xF5");
// GroupNames label

G:=SmallGroup(480,1197);
// by ID

G=gap.SmallGroup(480,1197);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,1356,9414,1210]);
// Polycyclic

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

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