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## G = A4×C15order 180 = 22·32·5

### Direct product of C15 and A4

Aliases: A4×C15, (C2×C30)⋊C3, (C2×C6)⋊C15, (C2×C10)⋊C32, C22⋊(C3×C15), SmallGroup(180,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C15
 Chief series C1 — C22 — C2×C10 — C5×A4 — A4×C15
 Lower central C22 — A4×C15
 Upper central C1 — C15

Generators and relations for A4×C15
G = < a,b,c,d | a15=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C15
On 60 points
Generators in S60
(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)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 16)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 46)(41 47)(42 48)(43 49)(44 50)(45 51)
(16 39 60)(17 40 46)(18 41 47)(19 42 48)(20 43 49)(21 44 50)(22 45 51)(23 31 52)(24 32 53)(25 33 54)(26 34 55)(27 35 56)(28 36 57)(29 37 58)(30 38 59)

G:=sub<Sym(60)| (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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,16)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,46)(41,47)(42,48)(43,49)(44,50)(45,51), (16,39,60)(17,40,46)(18,41,47)(19,42,48)(20,43,49)(21,44,50)(22,45,51)(23,31,52)(24,32,53)(25,33,54)(26,34,55)(27,35,56)(28,36,57)(29,37,58)(30,38,59)>;

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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,16)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,46)(41,47)(42,48)(43,49)(44,50)(45,51), (16,39,60)(17,40,46)(18,41,47)(19,42,48)(20,43,49)(21,44,50)(22,45,51)(23,31,52)(24,32,53)(25,33,54)(26,34,55)(27,35,56)(28,36,57)(29,37,58)(30,38,59) );

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)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,31),(24,32),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,16),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,46),(41,47),(42,48),(43,49),(44,50),(45,51)], [(16,39,60),(17,40,46),(18,41,47),(19,42,48),(20,43,49),(21,44,50),(22,45,51),(23,31,52),(24,32,53),(25,33,54),(26,34,55),(27,35,56),(28,36,57),(29,37,58),(30,38,59)]])

A4×C15 is a maximal subgroup of   A4⋊D15

60 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 15A ··· 15H 15I ··· 15AF 30A ··· 30H order 1 2 3 3 3 ··· 3 5 5 5 5 6 6 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 size 1 3 1 1 4 ··· 4 1 1 1 1 3 3 3 3 3 3 1 ··· 1 4 ··· 4 3 ··· 3

60 irreducible representations

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

Matrix representation of A4×C15 in GL4(𝔽31) generated by

 16 0 0 0 0 25 0 0 0 0 25 0 0 0 0 25
,
 1 0 0 0 0 30 0 0 0 30 0 1 0 30 1 0
,
 1 0 0 0 0 0 1 30 0 1 0 30 0 0 0 30
,
 5 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(31))| [16,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,30,30,30,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,1,0,0,1,0,0,0,30,30,30],[5,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A4×C15 in GAP, Magma, Sage, TeX

A_4\times C_{15}
% in TeX

G:=Group("A4xC15");
// GroupNames label

G:=SmallGroup(180,31);
// by ID

G=gap.SmallGroup(180,31);
# by ID

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

G:=Group<a,b,c,d|a^15=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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