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## G = C2×C32⋊3F5order 360 = 23·32·5

### Direct product of C2 and C32⋊3F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C2×C32⋊3F5
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — C32⋊3F5 — C2×C32⋊3F5
 Lower central C3×C15 — C2×C32⋊3F5
 Upper central C1 — C2

Generators and relations for C2×C323F5
G = < a,b,c,d,e | a2=b3=c3=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 528 in 96 conjugacy classes, 45 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, C32, D5, C10, Dic3, C2×C6, C15, C3×C6, C3×C6, F5, D10, C2×Dic3, C3×D5, C30, C3⋊Dic3, C62, C2×F5, C3×C15, C3⋊F5, C6×D5, C2×C3⋊Dic3, C32×D5, C3×C30, C2×C3⋊F5, C323F5, D5×C3×C6, C2×C323F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C3⋊S3, F5, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C2×F5, C3⋊F5, C2×C3⋊Dic3, C2×C3⋊F5, C323F5, C2×C323F5

Smallest permutation representation of C2×C323F5
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 34 19)(2 35 20)(3 31 16)(4 32 17)(5 33 18)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)(46 76 61)(47 77 62)(48 78 63)(49 79 64)(50 80 65)(51 81 66)(52 82 67)(53 83 68)(54 84 69)(55 85 70)(56 86 71)(57 87 72)(58 88 73)(59 89 74)(60 90 75)
(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 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)
(2 3 5 4)(6 13 7 15)(8 12 10 11)(9 14)(16 33 17 35)(18 32 20 31)(19 34)(21 43 22 45)(23 42 25 41)(24 44)(26 38 27 40)(28 37 30 36)(29 39)(46 48 47 50)(51 58 52 60)(53 57 55 56)(54 59)(61 78 62 80)(63 77 65 76)(64 79)(66 88 67 90)(68 87 70 86)(69 89)(71 83 72 85)(73 82 75 81)(74 84)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - + + image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 F5 C2×F5 C3⋊F5 C2×C3⋊F5 kernel C2×C32⋊3F5 C32⋊3F5 D5×C3×C6 C32×D5 C3×C30 C6×D5 C3×D5 C3×D5 C30 C3×C6 C32 C6 C3 # reps 1 2 1 2 2 4 4 4 4 1 1 8 8

Matrix representation of C2×C323F5 in GL8(𝔽61)

 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 60 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0 1 0 0 0 0 0 60 0 0 1 0 0 0 0 60 0 0 0
,
 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 50 11 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,50,50,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C2×C323F5 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_3F_5
% in TeX

G:=Group("C2xC3^2:3F5");
// GroupNames label

G:=SmallGroup(360,147);
// by ID

G=gap.SmallGroup(360,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,387,1444,7781,2609]);
// Polycyclic

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

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