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## G = C5×C3⋊S4order 360 = 23·32·5

### Direct product of C5 and C3⋊S4

Aliases: C5×C3⋊S4, C153S4, C3⋊(C5×S4), A4⋊(C5×S3), (C5×A4)⋊3S3, (C2×C30)⋊4S3, (A4×C15)⋊6C2, (C3×A4)⋊2C10, C22⋊(C5×C3⋊S3), (C2×C6)⋊2(C5×S3), (C2×C10)⋊1(C3⋊S3), SmallGroup(360,140)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C5×C3⋊S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — A4×C15 — C5×C3⋊S4
 Lower central C3×A4 — C5×C3⋊S4
 Upper central C1 — C5

Generators and relations for C5×C3⋊S4
G = < a,b,c,d,e,f | a5=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 280 in 60 conjugacy classes, 18 normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C22, C5, S3, C6, D4, C32, C10, Dic3, A4, D6, C2×C6, C15, C15, C3⋊S3, C20, C2×C10, C2×C10, C3⋊D4, S4, C5×S3, C30, C3×A4, C5×D4, C3×C15, C5×Dic3, C5×A4, S3×C10, C2×C30, C3⋊S4, C5×C3⋊S3, C5×C3⋊D4, C5×S4, A4×C15, C5×C3⋊S4
Quotients: C1, C2, C5, S3, C10, C3⋊S3, S4, C5×S3, C3⋊S4, C5×C3⋊S3, C5×S4, C5×C3⋊S4

Smallest permutation representation of C5×C3⋊S4
On 60 points
Generators in S60
(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 23 32)(2 24 33)(3 25 34)(4 21 35)(5 22 31)(6 58 48)(7 59 49)(8 60 50)(9 56 46)(10 57 47)(11 19 51)(12 20 52)(13 16 53)(14 17 54)(15 18 55)(26 45 36)(27 41 37)(28 42 38)(29 43 39)(30 44 40)
(1 20)(2 16)(3 17)(4 18)(5 19)(6 45)(7 41)(8 42)(9 43)(10 44)(11 31)(12 32)(13 33)(14 34)(15 35)(21 55)(22 51)(23 52)(24 53)(25 54)(26 48)(27 49)(28 50)(29 46)(30 47)(36 58)(37 59)(38 60)(39 56)(40 57)
(1 58)(2 59)(3 60)(4 56)(5 57)(6 32)(7 33)(8 34)(9 35)(10 31)(11 44)(12 45)(13 41)(14 42)(15 43)(16 37)(17 38)(18 39)(19 40)(20 36)(21 46)(22 47)(23 48)(24 49)(25 50)(26 52)(27 53)(28 54)(29 55)(30 51)
(1 23 32)(2 24 33)(3 25 34)(4 21 35)(5 22 31)(6 36 52)(7 37 53)(8 38 54)(9 39 55)(10 40 51)(11 57 30)(12 58 26)(13 59 27)(14 60 28)(15 56 29)(16 49 41)(17 50 42)(18 46 43)(19 47 44)(20 48 45)
(6 48)(7 49)(8 50)(9 46)(10 47)(11 30)(12 26)(13 27)(14 28)(15 29)(16 37)(17 38)(18 39)(19 40)(20 36)(21 35)(22 31)(23 32)(24 33)(25 34)(41 53)(42 54)(43 55)(44 51)(45 52)

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

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

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

45 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4 5A 5B 5C 5D 6 10A 10B 10C 10D 10E 10F 10G 10H 15A 15B 15C 15D 15E ··· 15P 20A 20B 20C 20D 30A 30B 30C 30D order 1 2 2 3 3 3 3 4 5 5 5 5 6 10 10 10 10 10 10 10 10 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 size 1 3 18 2 8 8 8 18 1 1 1 1 6 3 3 3 3 18 18 18 18 2 2 2 2 8 ··· 8 18 18 18 18 6 6 6 6

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + image C1 C2 C5 C10 S3 S3 C5×S3 C5×S3 S4 C5×S4 C3⋊S4 C5×C3⋊S4 kernel C5×C3⋊S4 A4×C15 C3⋊S4 C3×A4 C5×A4 C2×C30 A4 C2×C6 C15 C3 C5 C1 # reps 1 1 4 4 3 1 12 4 2 8 1 4

Matrix representation of C5×C3⋊S4 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 20 0 0 0 0 0 20 0 0 0 0 0 20
,
 60 60 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 0 1 0 0 0 0 0 0 0 1 0 0 1 60 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 60 0 0 0 0 60 1 60
,
 0 1 0 0 0 60 60 0 0 0 0 0 0 60 0 0 0 1 60 1 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 60

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,20],[60,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,60,0,0,0,1,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,60,0,1,0,0,0,0,60],[0,60,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,60,60,0,0,0,0,1,1],[0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,60] >;

C5×C3⋊S4 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes S_4
% in TeX

G:=Group("C5xC3:S4");
// GroupNames label

G:=SmallGroup(360,140);
// by ID

G=gap.SmallGroup(360,140);
# by ID

G:=PCGroup([6,-2,-5,-3,-3,-2,2,362,1443,5404,556,3245,989]);
// Polycyclic

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

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