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## G = C3×C3⋊D20order 360 = 23·32·5

### Direct product of C3 and C3⋊D20

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 2A 2B 2C 3A 3B 3C 3D 3E 4 5A 5B 6A 6B 6C 6D 6E 6F ··· 6M 6N 6O 10A 10B 12A 12B 15A 15B 15C 15D 15E ··· 15J 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30J 60A ··· 60H order 1 2 2 2 3 3 3 3 3 4 5 5 6 6 6 6 6 6 ··· 6 6 6 10 10 12 12 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 10 30 1 1 2 2 2 6 2 2 1 1 2 2 2 10 ··· 10 30 30 2 2 6 6 2 2 2 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D5 D6 C3×S3 D10 C3⋊D4 C3×D4 C3×D5 S3×C6 D20 C6×D5 C3×C3⋊D4 C3×D20 S3×D5 C3⋊D20 C3×S3×D5 C3×C3⋊D20 kernel C3×C3⋊D20 Dic3×C15 D5×C3×C6 C6×D15 C3⋊D20 C5×Dic3 C6×D5 D30 C6×D5 C3×C15 C3×Dic3 C30 D10 C3×C6 C15 C15 Dic3 C10 C32 C6 C5 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 2 1 2 2 2 2 4 2 4 4 4 8 2 2 4 4

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

 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 60 0 0 1 17 0 0 0 0 0 1 0 0 60 0
,
 44 60 0 0 44 17 0 0 0 0 0 1 0 0 1 0
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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