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G = S3×C4⋊F5  order 480 = 25·3·5

Direct product of S3 and C4⋊F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×C4⋊F5, C4⋊3(S3×F5), C20⋊3(C4×S3), D15⋊(C4⋊C4), (S3×D5).Q8, C60⋊3(C2×C4), (C4×S3)⋊2F5, C12⋊1(C2×F5), (S3×C20)⋊2C4, (C4×D15)⋊2C4, (S3×D5).1D4, D5.1(S3×D4), C60⋊C4⋊4C2, (C2×F5).3D6, D5.2(S3×Q8), Dic3⋊F5⋊1C2, D30.C2⋊7C4, Dic3⋊3(C2×F5), (S3×Dic5)⋊7C4, (C4×D5).67D6, D6.14(C2×F5), Dic15⋊4(C2×C4), Dic5⋊10(C4×S3), D30.12(C2×C4), (C6×F5).3C22, C6.15(C22×F5), C30.15(C22×C4), (C6×D5).27C23, (D5×C12).53C22, D10.30(C22×S3), (D5×Dic3).15C22, C5⋊(S3×C4⋊C4), C3⋊1(C2×C4⋊F5), C15⋊1(C2×C4⋊C4), (C2×S3×F5).C2, (C5×S3)⋊(C4⋊C4), (C3×C4⋊F5)⋊4C2, (C4×S3×D5).5C2, C2.18(C2×S3×F5), C10.15(S3×C2×C4), (C3×D5).2(C2×D4), (C3×D5).3(C2×Q8), (C2×C3⋊F5).4C22, (C5×Dic3)⋊4(C2×C4), (C3×Dic5)⋊9(C2×C4), (C2×S3×D5).17C22, (S3×C10).12(C2×C4), SmallGroup(480,996)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — S3×C4⋊F5
Chief series
C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — C2×S3×F5 — S3×C4⋊F5
Lower central
C15 — C30 — S3×C4⋊F5
Upper central
C1 — C2 — C4

Generators and relations for S3×C4⋊F5
 G = < a,b,c,d,e | a3=b2=c4=d5=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 1012 in 184 conjugacy classes, 60 normal (40 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, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×C4⋊C4, C4×D5, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, 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×C4⋊C4, 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, C3×C4⋊F5, C60⋊C4, C4×S3×D5, C2×S3×F5, S3×C4⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C4×S3, C22×S3, C2×C4⋊C4, C2×F5, S3×C2×C4, S3×D4, S3×Q8, C4⋊F5, C22×F5, S3×C4⋊C4, S3×F5, C2×C4⋊F5, C2×S3×F5, S3×C4⋊F5

Smallest permutation representation of S3×C4⋊F5
►On 60 points
Generators in S60
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(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 19)(2 20)(3 16)(4 17)(5 18)(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 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(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 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)

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C···4G4H···4L 5 6A6B6C10A10B10C12A12B···12F 15 20A20B20C20D 30 60A60B
order122222223444···44···456661010101212···121520202020306060
size113355151522610···1030···3042101041212420···208441212888

42 irreducible representations

dim111111111122222224444444888
type++++++++-+++++++-++
imageC1C2C2C2C2C2C4C4C4C4S3D4Q8D6D6C4×S3C4×S3F5C2×F5C2×F5C2×F5S3×D4S3×Q8C4⋊F5S3×F5C2×S3×F5S3×C4⋊F5
kernelS3×C4⋊F5Dic3⋊F5C3×C4⋊F5C60⋊C4C4×S3×D5C2×S3×F5S3×Dic5D30.C2S3×C20C4×D15C4⋊F5S3×D5S3×D5C4×D5C2×F5Dic5C20C4×S3Dic3C12D6D5D5S3C4C2C1
# reps121112222212212221111114112

Matrix representation of S3×C4⋊F5 ►in GL8(𝔽61)

6060000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
6060000000
00100000
00010000
000060000
000006000
000000600
000000060
,
600000000
060000000
000600000
00100000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000060100
000060010
000060001
000060000
,
10000000
01000000
005000000
000110000
00000010
00001000
00000001
00000100

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

S3×C4⋊F5 in GAP, Magma, Sage, TeX 

S_3\times C_4\rtimes F_5
 
% in TeX
 
G:=Group("S3xC4:F5");
 
// GroupNames label
 
G:=SmallGroup(480,996);
 
// by ID
 
G=gap.SmallGroup(480,996);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,100,1356,9414,2379]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^5=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
 
// generators/relations
 

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