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## G = C3×C5⋊S4order 360 = 23·32·5

### Direct product of C3 and C5⋊S4

Aliases: C3×C5⋊S4, C152S4, C5⋊(C3×S4), A4⋊(C3×D5), (C5×A4)⋊1C6, (C2×C30)⋊3S3, (C3×A4)⋊1D5, (C2×C6)⋊1D15, C22⋊(C3×D15), (A4×C15)⋊4C2, (C2×C10)⋊2(C3×S3), SmallGroup(360,139)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C3×C5⋊S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — A4×C15 — C3×C5⋊S4
 Lower central C5×A4 — C3×C5⋊S4
 Upper central C1 — C3

Generators and relations for C3×C5⋊S4
G = < a,b,c,d,e,f | a3=b5=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 >

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

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

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

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

39 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4 5A 5B 6A 6B 6C 6D 10A 10B 12A 12B 15A 15B 15C 15D 15E ··· 15P 30A 30B 30C 30D order 1 2 2 3 3 3 3 3 4 5 5 6 6 6 6 10 10 12 12 15 15 15 15 15 ··· 15 30 30 30 30 size 1 3 30 1 1 8 8 8 30 2 2 3 3 30 30 6 6 30 30 2 2 2 2 8 ··· 8 6 6 6 6

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 6 6 type + + + + + + + image C1 C2 C3 C6 S3 D5 C3×S3 C3×D5 D15 C3×D15 S4 C3×S4 C5⋊S4 C3×C5⋊S4 kernel C3×C5⋊S4 A4×C15 C5⋊S4 C5×A4 C2×C30 C3×A4 C2×C10 A4 C2×C6 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 2 2 4 4 8 2 4 2 4

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

 1 0 0 0 0 0 1 0 0 0 0 0 47 0 0 0 0 0 47 0 0 0 0 0 47
,
 31 42 0 0 0 19 12 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 0 0 0 1 0 0 0 0 60 60 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 60 60 0 0 0 0 1 0 0 0 1 0
,
 60 1 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 60 60 60
,
 31 42 0 0 0 12 30 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47],[31,19,0,0,0,42,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,1,0,60,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,60,0,1,0,0,60,1,0],[60,60,0,0,0,1,0,0,0,0,0,0,1,0,60,0,0,0,0,60,0,0,0,1,60],[31,12,0,0,0,42,30,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×C5⋊S4 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes S_4
% in TeX

G:=Group("C3xC5:S4");
// GroupNames label

G:=SmallGroup(360,139);
// by ID

G=gap.SmallGroup(360,139);
# by ID

G:=PCGroup([6,-2,-3,-3,-5,-2,2,218,1731,5404,916,3245,1637]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^5=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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