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## G = C3×C15⋊D4order 360 = 23·32·5

### Direct product of C3 and C15⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C3×C15⋊D4
 Chief series C1 — C5 — C15 — C30 — C3×C30 — D5×C3×C6 — C3×C15⋊D4
 Lower central C15 — C30 — C3×C15⋊D4
 Upper central C1 — C6

Generators and relations for C3×C15⋊D4
G = < a,b,c,d | a3=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b4, dcd=c-1 >

Subgroups: 276 in 74 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C3×C6, Dic5, D10, C2×C10, C3⋊D4, C3×D4, C5×S3, C3×D5, C30, C30, C3×Dic3, S3×C6, C62, C5⋊D4, C3×C15, C3×Dic5, Dic15, C6×D5, C6×D5, S3×C10, C2×C30, C3×C3⋊D4, C32×D5, S3×C15, C3×C30, C15⋊D4, C3×C5⋊D4, C3×Dic15, D5×C3×C6, S3×C30, C3×C15⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, D5, D6, C2×C6, C3×S3, D10, C3⋊D4, C3×D4, C3×D5, S3×C6, C5⋊D4, S3×D5, C6×D5, C3×C3⋊D4, C15⋊D4, C3×C5⋊D4, C3×S3×D5, C3×C15⋊D4

Smallest permutation representation of C3×C15⋊D4
On 60 points
Generators in S60
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(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 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)
(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 38 16 55)(2 37 17 54)(3 36 18 53)(4 35 19 52)(5 34 20 51)(6 33 21 50)(7 32 22 49)(8 31 23 48)(9 45 24 47)(10 44 25 46)(11 43 26 60)(12 42 27 59)(13 41 28 58)(14 40 29 57)(15 39 30 56)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(17 20)(18 24)(19 28)(22 25)(23 29)(27 30)(31 57)(32 46)(33 50)(34 54)(35 58)(36 47)(37 51)(38 55)(39 59)(40 48)(41 52)(42 56)(43 60)(44 49)(45 53)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H ··· 6O 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 15C 15D 15E ··· 15J 30A 30B 30C 30D 30E ··· 30J 30K ··· 30R order 1 2 2 2 3 3 3 3 3 4 5 5 6 6 6 6 6 6 6 6 ··· 6 10 10 10 10 10 10 12 12 15 15 15 15 15 ··· 15 30 30 30 30 30 ··· 30 30 ··· 30 size 1 1 6 10 1 1 2 2 2 30 2 2 1 1 2 2 2 6 6 10 ··· 10 2 2 6 6 6 6 30 30 2 2 2 2 4 ··· 4 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 C5⋊D4 C6×D5 C3×C3⋊D4 C3×C5⋊D4 S3×D5 C15⋊D4 C3×S3×D5 C3×C15⋊D4 kernel C3×C15⋊D4 C3×Dic15 D5×C3×C6 S3×C30 C15⋊D4 Dic15 C6×D5 S3×C10 C6×D5 C3×C15 S3×C6 C30 D10 C3×C6 C15 C15 D6 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×C15⋊D4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 47 0 0 0 0 47
,
 44 44 0 0 17 60 0 0 0 0 13 0 0 0 20 47
,
 14 45 0 0 39 47 0 0 0 0 60 41 0 0 55 1
,
 1 0 0 0 17 60 0 0 0 0 1 0 0 0 6 60
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[44,17,0,0,44,60,0,0,0,0,13,20,0,0,0,47],[14,39,0,0,45,47,0,0,0,0,60,55,0,0,41,1],[1,17,0,0,0,60,0,0,0,0,1,6,0,0,0,60] >;

C3×C15⋊D4 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes D_4
% in TeX

G:=Group("C3xC15:D4");
// GroupNames label

G:=SmallGroup(360,61);
// by ID

G=gap.SmallGroup(360,61);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,730,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^15=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^4,d*c*d=c^-1>;
// generators/relations

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