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G = C15×C3⋊D4order 360 = 23·32·5

Direct product of C15 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C15×C3⋊D4, D62C30, Dic3⋊C30, C622C10, C30.71D6, (C2×C30)⋊5S3, (C2×C6)⋊4C30, (C6×C30)⋊8C2, C32(D4×C15), C159(C3×D4), (C3×C15)⋊25D4, (S3×C6)⋊4C10, (S3×C10)⋊5C6, C326(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), C223(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

C1C6 — C15×C3⋊D4
C1C3C6C30C3×C30S3×C30 — C15×C3⋊D4
C3C6 — C15×C3⋊D4
C1C30C2×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 [×2], C3 [×2], C3, C4, C22, C22, C5, S3, C6 [×2], C6 [×6], D4, C32, C10, C10 [×2], Dic3, C12, D6, C2×C6 [×2], C2×C6 [×2], C15 [×2], C15, C3×S3, C3×C6, C3×C6, C20, C2×C10, C2×C10, C3⋊D4, C3×D4, C5×S3, C30 [×2], C30 [×6], C3×Dic3, S3×C6, C62, C5×D4, C3×C15, C5×Dic3, C60, S3×C10, C2×C30 [×2], C2×C30 [×2], 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 [×3], C3, C22, C5, S3, C6 [×3], D4, C10 [×3], D6, C2×C6, C15, C3×S3, C2×C10, C3⋊D4, C3×D4, C5×S3, C30 [×3], 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 41 25)(2 60 42 26)(3 46 43 27)(4 47 44 28)(5 48 45 29)(6 49 31 30)(7 50 32 16)(8 51 33 17)(9 52 34 18)(10 53 35 19)(11 54 36 20)(12 55 37 21)(13 56 38 22)(14 57 39 23)(15 58 40 24)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 46)(44 47)(45 48)

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

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

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

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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