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## G = Dic15⋊S3order 360 = 23·32·5

### 3rd semidirect product of Dic15 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — Dic15⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C3×Dic15 — Dic15⋊S3
 Lower central C3×C15 — Dic15⋊S3
 Upper central C1 — C2

Generators and relations for Dic15⋊S3
G = < a,b,c,d | a30=c3=d2=1, b2=a15, bab-1=a-1, ac=ca, dad=a11, bc=cb, bd=db, dcd=c-1 >

Subgroups: 348 in 74 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C15, C15, C3⋊S3, C3×C6, Dic5, C2×C10, C4×S3, C5×S3, C30, C30, C3×Dic3, C2×C3⋊S3, C2×Dic5, C3×C15, C3×Dic5, Dic15, S3×C10, C6.D6, C5×C3⋊S3, C3×C30, S3×Dic5, C3×Dic15, C10×C3⋊S3, Dic15⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, D6, Dic5, D10, C4×S3, S32, C2×Dic5, S3×D5, C6.D6, S3×Dic5, D15⋊S3, Dic15⋊S3

Smallest permutation representation of Dic15⋊S3
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 41 16 56)(2 40 17 55)(3 39 18 54)(4 38 19 53)(5 37 20 52)(6 36 21 51)(7 35 22 50)(8 34 23 49)(9 33 24 48)(10 32 25 47)(11 31 26 46)(12 60 27 45)(13 59 28 44)(14 58 29 43)(15 57 30 42)
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 28)(9 19 29)(10 20 30)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 51)(33 43)(34 54)(36 46)(37 57)(39 49)(40 60)(42 52)(45 55)(48 58)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 3 3 4 4 4 4 5 5 6 6 6 10 10 10 10 10 10 12 12 12 12 15 ··· 15 30 ··· 30 size 1 1 9 9 2 2 4 15 15 15 15 2 2 2 2 4 2 2 18 18 18 18 30 30 30 30 4 ··· 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + + + + - image C1 C2 C2 C4 S3 D5 D6 Dic5 D10 C4×S3 S32 S3×D5 C6.D6 S3×Dic5 D15⋊S3 Dic15⋊S3 kernel Dic15⋊S3 C3×Dic15 C10×C3⋊S3 C5×C3⋊S3 Dic15 C2×C3⋊S3 C30 C3⋊S3 C3×C6 C15 C10 C6 C5 C3 C2 C1 # reps 1 2 1 4 2 2 2 4 2 4 1 4 1 4 4 4

Matrix representation of Dic15⋊S3 in GL6(𝔽61)

 0 60 0 0 0 0 1 18 0 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 38 0 0 0 0 52 55 0 0 0 0 0 0 11 0 0 0 0 0 50 50 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 60 60

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,18,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,52,0,0,0,0,38,55,0,0,0,0,0,0,11,50,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60] >;

Dic15⋊S3 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes S_3
% in TeX

G:=Group("Dic15:S3");
// GroupNames label

G:=SmallGroup(360,85);
// by ID

G=gap.SmallGroup(360,85);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,387,201,730,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^3=d^2=1,b^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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