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## G = A4×D15order 360 = 23·32·5

### Direct product of A4 and D15

Aliases: A4×D15, C5⋊(S3×A4), C3⋊(D5×A4), (C5×A4)⋊2S3, C151(C2×A4), (C2×C30)⋊1C6, (C3×A4)⋊3D5, (A4×C15)⋊3C2, (C22×D15)⋊C3, C222(C3×D15), (C2×C6)⋊(C3×D5), (C2×C10)⋊3(C3×S3), SmallGroup(360,144)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — A4×D15
 Chief series C1 — C5 — C15 — C2×C30 — A4×C15 — A4×D15
 Lower central C2×C30 — A4×D15
 Upper central C1

Generators and relations for A4×D15
G = < a,b,c,d,e | a2=b2=c3=d15=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

3C2
15C2
45C2
4C3
8C3
45C22
45C22
3C6
5S3
15S3
60C6
4C32
3C10
3D5
9D5
4C15
8C15
15C23
2A4
15D6
15D6
20C3×S3
9D10
9D10
3C30
3D15
12C3×D5
15C2×A4
3D30
3D30

Smallest permutation representation of A4×D15
On 60 points
Generators in S60
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)
(16 38 53)(17 39 54)(18 40 55)(19 41 56)(20 42 57)(21 43 58)(22 44 59)(23 45 60)(24 31 46)(25 32 47)(26 33 48)(27 34 49)(28 35 50)(29 36 51)(30 37 52)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 5A 5B 6A 6B 6C 10A 10B 15A 15B 15C 15D 15E ··· 15P 30A 30B 30C 30D order 1 2 2 2 3 3 3 3 3 5 5 6 6 6 10 10 15 15 15 15 15 ··· 15 30 30 30 30 size 1 3 15 45 2 4 4 8 8 2 2 6 60 60 6 6 2 2 2 2 8 ··· 8 6 6 6 6

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 6 6 6 type + + + + + + + + + + image C1 C2 C3 C6 S3 D5 C3×S3 D15 C3×D5 C3×D15 A4 C2×A4 S3×A4 D5×A4 A4×D15 kernel A4×D15 A4×C15 C22×D15 C2×C30 C5×A4 C3×A4 C2×C10 A4 C2×C6 C22 D15 C15 C5 C3 C1 # reps 1 1 2 2 1 2 2 4 4 8 1 1 1 2 4

Matrix representation of A4×D15 in GL5(𝔽31)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 30 30 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 30 30 30 0 0 1 0 0
,
 5 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 30 30 30 0 0 0 1 0
,
 11 9 0 0 0 6 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 14 20 0 0 0 29 17 0 0 0 0 0 30 0 0 0 0 0 30 0 0 0 0 0 30

G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,30,0,0,1,0,30,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,0,30,1,0,0,0,30,0,0,0,1,30,0],[5,0,0,0,0,0,5,0,0,0,0,0,1,30,0,0,0,0,30,1,0,0,0,30,0],[11,6,0,0,0,9,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[14,29,0,0,0,20,17,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30] >;

A4×D15 in GAP, Magma, Sage, TeX

A_4\times D_{15}
% in TeX

G:=Group("A4xD15");
// GroupNames label

G:=SmallGroup(360,144);
// by ID

G=gap.SmallGroup(360,144);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-5,170,81,1444,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^15=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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