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G = C15×C3⋊D4  order 360 = 23·32·5

Direct product of C15 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C15×C3⋊D4, D6⋊2C30, Dic3⋊C30, C62⋊2C10, C30.71D6, (C2×C30)⋊5S3, (C2×C6)⋊4C30, (C6×C30)⋊8C2, C3⋊2(D4×C15), C15⋊9(C3×D4), (C3×C15)⋊25D4, (S3×C6)⋊4C10, (S3×C10)⋊5C6, C32⋊6(C5×D4), (C2×C30)⋊10C6, C2.5(S3×C30), C6.5(C2×C30), (S3×C30)⋊10C2, C10.17(S3×C6), C6.21(S3×C10), C30.28(C2×C6), C22⋊3(S3×C15), (C5×Dic3)⋊4C6, (C3×Dic3)⋊4C10, (Dic3×C15)⋊10C2, (C3×C30).51C22, (C2×C6)⋊3(C5×S3), (C2×C10)⋊5(C3×S3), (C3×C6).10(C2×C10), SmallGroup(360,99)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C15×C3⋊D4
Chief series
C1 — C3 — C6 — C30 — C3×C30 — S3×C30 — C15×C3⋊D4
Lower central
C3 — C6 — C15×C3⋊D4
Upper central
C1 — C30 — C2×C30

Generators and relations for C15×C3⋊D4
 G = < a,b,c,d | a15=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 140 in 74 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C5, S3, C6, C6, D4, C32, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C15, C3×S3, C3×C6, C3×C6, C20, C2×C10, C2×C10, C3⋊D4, C3×D4, C5×S3, C30, C30, C3×Dic3, S3×C6, C62, C5×D4, C3×C15, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C3×C3⋊D4, S3×C15, C3×C30, C3×C30, C5×C3⋊D4, D4×C15, Dic3×C15, S3×C30, C6×C30, C15×C3⋊D4
Quotients: C1, C2, C3, C22, C5, S3, C6, D4, C10, D6, C2×C6, C15, C3×S3, C2×C10, C3⋊D4, C3×D4, C5×S3, C30, S3×C6, C5×D4, S3×C10, C2×C30, C3×C3⋊D4, S3×C15, C5×C3⋊D4, D4×C15, S3×C30, C15×C3⋊D4

Smallest permutation representation of C15×C3⋊D4
►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 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)

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

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

135 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 5A5B5C5D6A6B6C···6M6N6O10A10B10C10D10E10F10G10H10I10J10K10L12A12B15A···15H15I···15T20A20B20C20D30A···30H30I···30AZ30BA···30BH60A···60H
order12223333345555666···666101010101010101010101010121215···1515···152020202030···3030···3030···3060···60
size11261122261111112···266111122226666661···12···266661···12···26···66···6

135 irreducible representations

dim11111111111111112222222222222222
type+++++++
imageC1C2C2C2C3C5C6C6C6C10C10C10C15C30C30C30S3D4D6C3×S3C3⋊D4C3×D4C5×S3S3×C6C5×D4S3×C10C3×C3⋊D4S3×C15C5×C3⋊D4D4×C15S3×C30C15×C3⋊D4
kernelC15×C3⋊D4Dic3×C15S3×C30C6×C30C5×C3⋊D4C3×C3⋊D4C5×Dic3S3×C10C2×C30C3×Dic3S3×C6C62C3⋊D4Dic3D6C2×C6C2×C30C3×C15C30C2×C10C15C15C2×C6C10C32C6C5C22C3C3C2C1
# reps111124222444888811122242444888816

Matrix representation of C15×C3⋊D4 ►in GL2(𝔽31) generated by

280
028
,
113
2629
,
204
1611
,
3018
01
G:=sub<GL(2,GF(31))| [28,0,0,28],[1,26,13,29],[20,16,4,11],[30,0,18,1] >;
 

C15×C3⋊D4 in GAP, Magma, Sage, TeX 

C_{15}\times C_3\rtimes D_4
 
% in TeX
 
G:=Group("C15xC3:D4");
 
// GroupNames label
 
G:=SmallGroup(360,99);
 
// by ID
 
G=gap.SmallGroup(360,99);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-5,-2,-3,745,8645]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^15=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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