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G = C3×C3⋊D20  order 360 = 23·32·5

Direct product of C3 and C3⋊D20

direct product, metabelian, supersoluble, monomial

Aliases: C3×C3⋊D20, D30⋊3C6, C32⋊6D20, C30.32D6, (C3×C15)⋊5D4, (C6×D5)⋊5S3, (C6×D5)⋊2C6, C15⋊2(C3×D4), C3⋊2(C3×D20), Dic3⋊(C3×D5), C6.5(C6×D5), (C6×D15)⋊3C2, D10⋊2(C3×S3), C10.5(S3×C6), C30.5(C2×C6), C6.32(S3×D5), C15⋊8(C3⋊D4), (C5×Dic3)⋊3C6, (C3×Dic3)⋊4D5, (C3×C6).17D10, (Dic3×C15)⋊4C2, (C3×C30).5C22, (D5×C3×C6)⋊2C2, C5⋊1(C3×C3⋊D4), C2.5(C3×S3×D5), SmallGroup(360,62)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C3×C3⋊D20
Chief series
C1 — C5 — C15 — C30 — C3×C30 — D5×C3×C6 — C3×C3⋊D20
Lower central
C15 — C30 — C3×C3⋊D20
Upper central
C1 — C6

Generators and relations for C3×C3⋊D20
 G = < a,b,c,d | a3=b3=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 340 in 74 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C3×C6, C20, D10, D10, C3⋊D4, C3×D4, C3×D5, D15, C30, C30, C3×Dic3, S3×C6, C62, D20, C3×C15, C5×Dic3, C60, C6×D5, C6×D5, D30, C3×C3⋊D4, C32×D5, C3×D15, C3×C30, C3⋊D20, C3×D20, Dic3×C15, D5×C3×C6, C6×D15, C3×C3⋊D20
Quotients: C1, C2, C3, C22, S3, C6, D4, D5, D6, C2×C6, C3×S3, D10, C3⋊D4, C3×D4, C3×D5, S3×C6, D20, S3×D5, C6×D5, C3×C3⋊D4, C3⋊D20, C3×D20, C3×S3×D5, C3×C3⋊D20

Smallest permutation representation of C3×C3⋊D20
►On 60 points
Generators in S60
(1 45 29)(2 46 30)(3 47 31)(4 48 32)(5 49 33)(6 50 34)(7 51 35)(8 52 36)(9 53 37)(10 54 38)(11 55 39)(12 56 40)(13 57 21)(14 58 22)(15 59 23)(16 60 24)(17 41 25)(18 42 26)(19 43 27)(20 44 28)

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

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

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 5A5B6A6B6C6D6E6F···6M6N6O10A10B12A12B15A15B15C15D15E···15J20A20B20C20D30A30B30C30D30E···30J60A···60H
order122233333455666666···666101012121515151515···15202020203030303030···3060···60
size111030112226221122210···103030226622224···4666622224···46···6

63 irreducible representations

dim11111111222222222222224444
type++++++++++++
imageC1C2C2C2C3C6C6C6S3D4D5D6C3×S3D10C3⋊D4C3×D4C3×D5S3×C6D20C6×D5C3×C3⋊D4C3×D20S3×D5C3⋊D20C3×S3×D5C3×C3⋊D20
kernelC3×C3⋊D20Dic3×C15D5×C3×C6C6×D15C3⋊D20C5×Dic3C6×D5D30C6×D5C3×C15C3×Dic3C30D10C3×C6C15C15Dic3C10C32C6C5C3C6C3C2C1
# reps11112222112122224244482244

Matrix representation of C3×C3⋊D20 ►in GL4(𝔽61) generated by

1000
0100
00130
00013
,
1000
0100
00130
00047
,
06000
11700
0001
00600
,
446000
441700
0001
0010
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,47],[0,1,0,0,60,17,0,0,0,0,0,60,0,0,1,0],[44,44,0,0,60,17,0,0,0,0,0,1,0,0,1,0] >;
 

C3×C3⋊D20 in GAP, Magma, Sage, TeX 

C_3\times C_3\rtimes D_{20}
 
% in TeX
 
G:=Group("C3xC3:D20");
 
// GroupNames label
 
G:=SmallGroup(360,62);
 
// by ID
 
G=gap.SmallGroup(360,62);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,79,730,10373]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^20=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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