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## G = C5×C3.A4order 180 = 22·32·5

### Direct product of C5 and C3.A4

Aliases: C5×C3.A4, C15.A4, C22⋊C45, (C2×C10)⋊C9, C3.(C5×A4), (C2×C30).C3, (C2×C6).C15, SmallGroup(180,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C3.A4
 Chief series C1 — C22 — C2×C6 — C2×C30 — C5×C3.A4
 Lower central C22 — C5×C3.A4
 Upper central C1 — C15

Generators and relations for C5×C3.A4
G = < a,b,c,d,e | a5=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C5×C3.A4
On 90 points
Generators in S90
(1 21 57 76 11)(2 22 58 77 12)(3 23 59 78 13)(4 24 60 79 14)(5 25 61 80 15)(6 26 62 81 16)(7 27 63 73 17)(8 19 55 74 18)(9 20 56 75 10)(28 52 67 89 37)(29 53 68 90 38)(30 54 69 82 39)(31 46 70 83 40)(32 47 71 84 41)(33 48 72 85 42)(34 49 64 86 43)(35 50 65 87 44)(36 51 66 88 45)
(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 49)(3 51)(4 52)(6 54)(7 46)(9 48)(10 33)(11 34)(13 36)(14 28)(16 30)(17 31)(20 72)(21 64)(23 66)(24 67)(26 69)(27 70)(37 79)(39 81)(40 73)(42 75)(43 76)(45 78)(56 85)(57 86)(59 88)(60 89)(62 82)(63 83)
(1 49)(2 50)(4 52)(5 53)(7 46)(8 47)(11 34)(12 35)(14 28)(15 29)(17 31)(18 32)(19 71)(21 64)(22 65)(24 67)(25 68)(27 70)(37 79)(38 80)(40 73)(41 74)(43 76)(44 77)(55 84)(57 86)(58 87)(60 89)(61 90)(63 83)
(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)

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

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

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

C5×C3.A4 is a maximal subgroup of   C22⋊D45

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C3 C5 C9 C15 C45 A4 C3.A4 C5×A4 C5×C3.A4 kernel C5×C3.A4 C2×C30 C3.A4 C2×C10 C2×C6 C22 C15 C5 C3 C1 # reps 1 2 4 6 8 24 1 2 4 8

Matrix representation of C5×C3.A4 in GL3(𝔽181) generated by

 135 0 0 0 135 0 0 0 135
,
 48 0 0 0 48 0 0 0 48
,
 180 0 0 0 180 0 112 0 1
,
 180 0 0 7 1 0 0 0 180
,
 174 179 0 171 7 1 6 112 0
G:=sub<GL(3,GF(181))| [135,0,0,0,135,0,0,0,135],[48,0,0,0,48,0,0,0,48],[180,0,112,0,180,0,0,0,1],[180,7,0,0,1,0,0,0,180],[174,171,6,179,7,112,0,1,0] >;

C5×C3.A4 in GAP, Magma, Sage, TeX

C_5\times C_3.A_4
% in TeX

G:=Group("C5xC3.A4");
// GroupNames label

G:=SmallGroup(180,8);
// by ID

G=gap.SmallGroup(180,8);
# by ID

G:=PCGroup([5,-3,-5,-3,-2,2,75,1803,3379]);
// Polycyclic

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

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