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## G = C5×C3.S4order 360 = 23·32·5

### Direct product of C5 and C3.S4

Aliases: C5×C3.S4, C15.2S4, C3.(C5×S4), C3.A4⋊C10, C22⋊(C5×D9), (C2×C10)⋊1D9, (C2×C30).2S3, (C2×C6).(C5×S3), (C5×C3.A4)⋊2C2, SmallGroup(360,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4 — C5×C3.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C5×C3.A4 — 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=f2=1, e3=b, 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=b-1e2 >

Smallest permutation representation of C5×C3.S4
On 90 points
Generators in S90
(1 88 70 52 34)(2 89 71 53 35)(3 90 72 54 36)(4 82 64 46 28)(5 83 65 47 29)(6 84 66 48 30)(7 85 67 49 31)(8 86 68 50 32)(9 87 69 51 33)(10 75 57 39 21)(11 76 58 40 22)(12 77 59 41 23)(13 78 60 42 24)(14 79 61 43 25)(15 80 62 44 26)(16 81 63 45 27)(17 73 55 37 19)(18 74 56 38 20)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)
(1 22)(2 23)(4 25)(5 26)(7 19)(8 20)(11 88)(12 89)(14 82)(15 83)(17 85)(18 86)(28 43)(29 44)(31 37)(32 38)(34 40)(35 41)(46 61)(47 62)(49 55)(50 56)(52 58)(53 59)(64 79)(65 80)(67 73)(68 74)(70 76)(71 77)
(2 23)(3 24)(5 26)(6 27)(8 20)(9 21)(10 87)(12 89)(13 90)(15 83)(16 84)(18 86)(29 44)(30 45)(32 38)(33 39)(35 41)(36 42)(47 62)(48 63)(50 56)(51 57)(53 59)(54 60)(65 80)(66 81)(68 74)(69 75)(71 77)(72 78)
(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 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)
(1 22)(2 21)(3 20)(4 19)(5 27)(6 26)(7 25)(8 24)(9 23)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 90)(28 37)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 55)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(64 73)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)

G:=sub<Sym(90)| (1,88,70,52,34)(2,89,71,53,35)(3,90,72,54,36)(4,82,64,46,28)(5,83,65,47,29)(6,84,66,48,30)(7,85,67,49,31)(8,86,68,50,32)(9,87,69,51,33)(10,75,57,39,21)(11,76,58,40,22)(12,77,59,41,23)(13,78,60,42,24)(14,79,61,43,25)(15,80,62,44,26)(16,81,63,45,27)(17,73,55,37,19)(18,74,56,38,20), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90), (1,22)(2,23)(4,25)(5,26)(7,19)(8,20)(11,88)(12,89)(14,82)(15,83)(17,85)(18,86)(28,43)(29,44)(31,37)(32,38)(34,40)(35,41)(46,61)(47,62)(49,55)(50,56)(52,58)(53,59)(64,79)(65,80)(67,73)(68,74)(70,76)(71,77), (2,23)(3,24)(5,26)(6,27)(8,20)(9,21)(10,87)(12,89)(13,90)(15,83)(16,84)(18,86)(29,44)(30,45)(32,38)(33,39)(35,41)(36,42)(47,62)(48,63)(50,56)(51,57)(53,59)(54,60)(65,80)(66,81)(68,74)(69,75)(71,77)(72,78), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,90)(28,37)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,55)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(64,73)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)>;

G:=Group( (1,88,70,52,34)(2,89,71,53,35)(3,90,72,54,36)(4,82,64,46,28)(5,83,65,47,29)(6,84,66,48,30)(7,85,67,49,31)(8,86,68,50,32)(9,87,69,51,33)(10,75,57,39,21)(11,76,58,40,22)(12,77,59,41,23)(13,78,60,42,24)(14,79,61,43,25)(15,80,62,44,26)(16,81,63,45,27)(17,73,55,37,19)(18,74,56,38,20), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90), (1,22)(2,23)(4,25)(5,26)(7,19)(8,20)(11,88)(12,89)(14,82)(15,83)(17,85)(18,86)(28,43)(29,44)(31,37)(32,38)(34,40)(35,41)(46,61)(47,62)(49,55)(50,56)(52,58)(53,59)(64,79)(65,80)(67,73)(68,74)(70,76)(71,77), (2,23)(3,24)(5,26)(6,27)(8,20)(9,21)(10,87)(12,89)(13,90)(15,83)(16,84)(18,86)(29,44)(30,45)(32,38)(33,39)(35,41)(36,42)(47,62)(48,63)(50,56)(51,57)(53,59)(54,60)(65,80)(66,81)(68,74)(69,75)(71,77)(72,78), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,90)(28,37)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,55)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(64,73)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74) );

G=PermutationGroup([[(1,88,70,52,34),(2,89,71,53,35),(3,90,72,54,36),(4,82,64,46,28),(5,83,65,47,29),(6,84,66,48,30),(7,85,67,49,31),(8,86,68,50,32),(9,87,69,51,33),(10,75,57,39,21),(11,76,58,40,22),(12,77,59,41,23),(13,78,60,42,24),(14,79,61,43,25),(15,80,62,44,26),(16,81,63,45,27),(17,73,55,37,19),(18,74,56,38,20)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90)], [(1,22),(2,23),(4,25),(5,26),(7,19),(8,20),(11,88),(12,89),(14,82),(15,83),(17,85),(18,86),(28,43),(29,44),(31,37),(32,38),(34,40),(35,41),(46,61),(47,62),(49,55),(50,56),(52,58),(53,59),(64,79),(65,80),(67,73),(68,74),(70,76),(71,77)], [(2,23),(3,24),(5,26),(6,27),(8,20),(9,21),(10,87),(12,89),(13,90),(15,83),(16,84),(18,86),(29,44),(30,45),(32,38),(33,39),(35,41),(36,42),(47,62),(48,63),(50,56),(51,57),(53,59),(54,60),(65,80),(66,81),(68,74),(69,75),(71,77),(72,78)], [(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90)], [(1,22),(2,21),(3,20),(4,19),(5,27),(6,26),(7,25),(8,24),(9,23),(10,89),(11,88),(12,87),(13,86),(14,85),(15,84),(16,83),(17,82),(18,90),(28,37),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,55),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(64,73),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74)]])

45 conjugacy classes

 class 1 2A 2B 3 4 5A 5B 5C 5D 6 9A 9B 9C 10A 10B 10C 10D 10E 10F 10G 10H 15A 15B 15C 15D 20A 20B 20C 20D 30A 30B 30C 30D 45A ··· 45L order 1 2 2 3 4 5 5 5 5 6 9 9 9 10 10 10 10 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 45 ··· 45 size 1 3 18 2 18 1 1 1 1 6 8 8 8 3 3 3 3 18 18 18 18 2 2 2 2 18 18 18 18 6 6 6 6 8 ··· 8

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + image C1 C2 C5 C10 S3 D9 C5×S3 C5×D9 S4 C5×S4 C3.S4 C5×C3.S4 kernel C5×C3.S4 C5×C3.A4 C3.S4 C3.A4 C2×C30 C2×C10 C2×C6 C22 C15 C3 C5 C1 # reps 1 1 4 4 1 3 4 12 2 8 1 4

Matrix representation of C5×C3.S4 in GL5(𝔽181)

 1 0 0 0 0 0 1 0 0 0 0 0 42 0 0 0 0 0 42 0 0 0 0 0 42
,
 180 180 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 0 180 0 0 0 0 0 1 0 0 0 0 180 180
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 180 0 0 0 180 0 180
,
 50 177 0 0 0 4 54 0 0 0 0 0 0 1 0 0 0 180 180 179 0 0 0 0 1
,
 1 0 0 0 0 180 180 0 0 0 0 0 1 0 0 0 0 180 180 179 0 0 0 0 1

G:=sub<GL(5,GF(181))| [1,0,0,0,0,0,1,0,0,0,0,0,42,0,0,0,0,0,42,0,0,0,0,0,42],[180,1,0,0,0,180,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,180,0,0,0,0,0,1,180,0,0,0,0,180],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,180,0,0,0,180,0,0,0,0,0,180],[50,4,0,0,0,177,54,0,0,0,0,0,0,180,0,0,0,1,180,0,0,0,0,179,1],[1,180,0,0,0,0,180,0,0,0,0,0,1,180,0,0,0,0,180,0,0,0,0,179,1] >;

C5×C3.S4 in GAP, Magma, Sage, TeX

C_5\times C_3.S_4
% in TeX

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

G:=SmallGroup(360,40);
// by ID

G=gap.SmallGroup(360,40);
# by ID

G:=PCGroup([6,-2,-5,-3,-3,-2,2,902,122,1443,5404,556,3245,989]);
// Polycyclic

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

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