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## G = A4×C30order 360 = 23·32·5

### Direct product of C30 and A4

Aliases: A4×C30, C23⋊(C3×C15), (C2×C6)⋊2C30, (C2×C30)⋊4C6, (C22×C6)⋊C15, (C22×C30)⋊C3, C22⋊(C3×C30), (C22×C10)⋊C32, (C2×C10)⋊2(C3×C6), SmallGroup(360,156)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C30
 Chief series C1 — C22 — C2×C10 — C2×C30 — A4×C15 — A4×C30
 Lower central C22 — A4×C30
 Upper central C1 — C30

Generators and relations for A4×C30
G = < a,b,c,d | a30=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Subgroups: 144 in 64 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C3, C22, C22, C5, C6, C6, C23, C32, C10, C10, A4, C2×C6, C2×C6, C15, C15, C3×C6, C2×C10, C2×C10, C2×A4, C22×C6, C30, C30, C3×A4, C22×C10, C3×C15, C5×A4, C2×C30, C2×C30, C6×A4, C3×C30, C10×A4, C22×C30, A4×C15, A4×C30
Quotients: C1, C2, C3, C5, C6, C32, C10, A4, C15, C3×C6, C2×A4, C30, C3×A4, C3×C15, C5×A4, C6×A4, C3×C30, C10×A4, A4×C15, A4×C30

Smallest permutation representation of A4×C30
On 90 points
Generators in S90
(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 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)
(1 83 35)(2 84 36)(3 85 37)(4 86 38)(5 87 39)(6 88 40)(7 89 41)(8 90 42)(9 61 43)(10 62 44)(11 63 45)(12 64 46)(13 65 47)(14 66 48)(15 67 49)(16 68 50)(17 69 51)(18 70 52)(19 71 53)(20 72 54)(21 73 55)(22 74 56)(23 75 57)(24 76 58)(25 77 59)(26 78 60)(27 79 31)(28 80 32)(29 81 33)(30 82 34)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G ··· 6L 10A 10B 10C 10D 10E ··· 10L 15A ··· 15H 15I ··· 15AF 30A ··· 30H 30I ··· 30X 30Y ··· 30AV order 1 2 2 2 3 3 3 ··· 3 5 5 5 5 6 6 6 6 6 6 6 ··· 6 10 10 10 10 10 ··· 10 15 ··· 15 15 ··· 15 30 ··· 30 30 ··· 30 30 ··· 30 size 1 1 3 3 1 1 4 ··· 4 1 1 1 1 1 1 3 3 3 3 4 ··· 4 1 1 1 1 3 ··· 3 1 ··· 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

120 irreducible representations

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

Matrix representation of A4×C30 in GL3(𝔽31) generated by

 12 0 0 0 12 0 0 0 12
,
 30 0 0 0 30 0 3 0 1
,
 30 0 0 26 1 0 0 0 30
,
 25 21 0 8 6 5 22 15 0
G:=sub<GL(3,GF(31))| [12,0,0,0,12,0,0,0,12],[30,0,3,0,30,0,0,0,1],[30,26,0,0,1,0,0,0,30],[25,8,22,21,6,15,0,5,0] >;

A4×C30 in GAP, Magma, Sage, TeX

A_4\times C_{30}
% in TeX

G:=Group("A4xC30");
// GroupNames label

G:=SmallGroup(360,156);
// by ID

G=gap.SmallGroup(360,156);
# by ID

G:=PCGroup([6,-2,-3,-3,-5,-2,2,2710,4871]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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