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## G = C2×D5×C3⋊S3order 360 = 23·32·5

### Direct product of C2, D5 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C2×D5×C3⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — D5×C3⋊S3 — C2×D5×C3⋊S3
 Lower central C3×C15 — C2×D5×C3⋊S3
 Upper central C1 — C2

Generators and relations for C2×D5×C3⋊S3
G = < a,b,c,d,e,f | a2=b5=c2=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1320 in 192 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C3, C22, C5, S3, C6, C6, C23, C32, D5, D5, C10, C10, D6, C2×C6, C15, C3⋊S3, C3⋊S3, C3×C6, C3×C6, D10, D10, C2×C10, C22×S3, C5×S3, C3×D5, D15, C30, C2×C3⋊S3, C2×C3⋊S3, C62, C22×D5, C3×C15, S3×D5, C6×D5, S3×C10, D30, C22×C3⋊S3, C32×D5, C5×C3⋊S3, C3⋊D15, C3×C30, C2×S3×D5, D5×C3⋊S3, D5×C3×C6, C10×C3⋊S3, C2×C3⋊D15, C2×D5×C3⋊S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C3⋊S3, D10, C22×S3, C2×C3⋊S3, C22×D5, S3×D5, C22×C3⋊S3, C2×S3×D5, D5×C3⋊S3, C2×D5×C3⋊S3

Smallest permutation representation of C2×D5×C3⋊S3
On 90 points
Generators in S90
(1 49)(2 50)(3 46)(4 47)(5 48)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)
(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 48)(2 47)(3 46)(4 50)(5 49)(6 51)(7 55)(8 54)(9 53)(10 52)(11 56)(12 60)(13 59)(14 58)(15 57)(16 61)(17 65)(18 64)(19 63)(20 62)(21 66)(22 70)(23 69)(24 68)(25 67)(26 71)(27 75)(28 74)(29 73)(30 72)(31 76)(32 80)(33 79)(34 78)(35 77)(36 81)(37 85)(38 84)(39 83)(40 82)(41 86)(42 90)(43 89)(44 88)(45 87)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)
(1 19 34)(2 20 35)(3 16 31)(4 17 32)(5 18 33)(6 21 36)(7 22 37)(8 23 38)(9 24 39)(10 25 40)(11 26 41)(12 27 42)(13 28 43)(14 29 44)(15 30 45)(46 61 76)(47 62 77)(48 63 78)(49 64 79)(50 65 80)(51 66 81)(52 67 82)(53 68 83)(54 69 84)(55 70 85)(56 71 86)(57 72 87)(58 73 88)(59 74 89)(60 75 90)
(1 49)(2 50)(3 46)(4 47)(5 48)(6 56)(7 57)(8 58)(9 59)(10 60)(11 51)(12 52)(13 53)(14 54)(15 55)(16 76)(17 77)(18 78)(19 79)(20 80)(21 86)(22 87)(23 88)(24 89)(25 90)(26 81)(27 82)(28 83)(29 84)(30 85)(31 61)(32 62)(33 63)(34 64)(35 65)(36 71)(37 72)(38 73)(39 74)(40 75)(41 66)(42 67)(43 68)(44 69)(45 70)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D5 D6 D6 D10 D10 S3×D5 C2×S3×D5 kernel C2×D5×C3⋊S3 D5×C3⋊S3 D5×C3×C6 C10×C3⋊S3 C2×C3⋊D15 C6×D5 C2×C3⋊S3 C3×D5 C30 C3⋊S3 C3×C6 C6 C3 # reps 1 4 1 1 1 4 2 8 4 4 2 8 8

Matrix representation of C2×D5×C3⋊S3 in GL7(𝔽31)

 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 1 0 0 0 0 0 11 19 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 30 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 30 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 26 0 0 0 0 0 13 29
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 30 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 5 0 0 0 0 0 0 1

G:=sub<GL(7,GF(31))| [30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,11,0,0,0,0,0,1,19,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,30,0,0,0,0,0,1,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,30,0,0,0,0,0,1,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,13,0,0,0,0,0,26,29],[1,0,0,0,0,0,0,0,1,30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,0,0,0,0,0,0,5,1] >;

C2×D5×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times D_5\times C_3\rtimes S_3
% in TeX

G:=Group("C2xD5xC3:S3");
// GroupNames label

G:=SmallGroup(360,152);
// by ID

G=gap.SmallGroup(360,152);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,201,730,10373]);
// Polycyclic

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

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