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## G = S3×C22⋊F5order 480 = 25·3·5

### Direct product of S3 and C22⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×C22⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — C2×S3×F5 — S3×C22⋊F5
 Lower central C15 — C30 — S3×C22⋊F5
 Upper central C1 — C2 — C22

Generators and relations for S3×C22⋊F5
G = < a,b,c,d,e,f | a3=b2=c2=d2=e5=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 1764 in 264 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, F5, D10, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C5×S3, C3×D5, C3×D5, D15, D15, C30, C30, C2×C22⋊C4, C2×F5, C2×F5, C22×D5, C22×D5, C22×C10, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C3×F5, C3⋊F5, S3×D5, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, C22⋊F5, C22⋊F5, C22×F5, C23×D5, S3×C22⋊C4, S3×F5, C6×F5, C2×C3⋊F5, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C2×C22⋊F5, D6⋊F5, C3×C22⋊F5, D10.D6, C2×S3×F5, C22×S3×D5, S3×C22⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, F5, C4×S3, C22×S3, C2×C22⋊C4, C2×F5, S3×C2×C4, S3×D4, C22⋊F5, C22×F5, S3×C22⋊C4, S3×F5, C2×C22⋊F5, C2×S3×F5, S3×C22⋊F5

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 6D 6E 10A 10B 10C 10D 10E 10F 10G 12A 12B 12C 12D 15 30A 30B 30C order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 6 6 6 6 6 10 10 10 10 10 10 10 12 12 12 12 15 30 30 30 size 1 1 2 3 3 5 5 6 10 15 15 30 2 10 10 10 10 30 30 30 30 4 2 4 10 10 20 4 4 4 12 12 12 12 20 20 20 20 8 8 8 8

42 irreducible representations

 dim 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 S3 D4 D6 D6 C4×S3 C4×S3 F5 C2×F5 C2×F5 S3×D4 C22⋊F5 S3×F5 C2×S3×F5 S3×C22⋊F5 kernel S3×C22⋊F5 D6⋊F5 C3×C22⋊F5 D10.D6 C2×S3×F5 C22×S3×D5 C2×S3×D5 S3×C2×C10 C22×D15 C22⋊F5 S3×D5 C2×F5 C22×D5 D10 C2×C10 C22×S3 D6 C2×C6 D5 S3 C22 C2 C1 # reps 1 2 1 1 2 1 4 2 2 1 4 2 1 2 2 1 2 1 2 4 1 1 2

Matrix representation of S3×C22⋊F5 in GL6(𝔽61)

 0 60 0 0 0 0 1 60 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
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 54 0 47 47 0 0 14 7 14 0 0 0 0 14 7 14 0 0 47 47 0 54
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 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 60 60 60 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 54 0 47 47 0 0 47 47 0 54 0 0 14 7 14 0 0 0 7 54 54 7

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,54,14,0,47,0,0,0,7,14,47,0,0,47,14,7,0,0,0,47,0,14,54],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,54,47,14,7,0,0,0,47,7,54,0,0,47,0,14,54,0,0,47,54,0,7] >;

S3×C22⋊F5 in GAP, Magma, Sage, TeX

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

G:=Group("S3xC2^2:F5");
// GroupNames label

G:=SmallGroup(480,1011);
// by ID

G=gap.SmallGroup(480,1011);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,1356,9414,2379]);
// Polycyclic

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

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