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## G = C15×C3⋊D4order 360 = 23·32·5

### Direct product of C15 and C3⋊D4

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 [×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 2A 2B 2C 3A 3B 3C 3D 3E 4 5A 5B 5C 5D 6A 6B 6C ··· 6M 6N 6O 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 15A ··· 15H 15I ··· 15T 20A 20B 20C 20D 30A ··· 30H 30I ··· 30AZ 30BA ··· 30BH 60A ··· 60H order 1 2 2 2 3 3 3 3 3 4 5 5 5 5 6 6 6 ··· 6 6 6 10 10 10 10 10 10 10 10 10 10 10 10 12 12 15 ··· 15 15 ··· 15 20 20 20 20 30 ··· 30 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 6 1 1 2 2 2 6 1 1 1 1 1 1 2 ··· 2 6 6 1 1 1 1 2 2 2 2 6 6 6 6 6 6 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 6 ··· 6

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C5 C6 C6 C6 C10 C10 C10 C15 C30 C30 C30 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 C5×S3 S3×C6 C5×D4 S3×C10 C3×C3⋊D4 S3×C15 C5×C3⋊D4 D4×C15 S3×C30 C15×C3⋊D4 kernel C15×C3⋊D4 Dic3×C15 S3×C30 C6×C30 C5×C3⋊D4 C3×C3⋊D4 C5×Dic3 S3×C10 C2×C30 C3×Dic3 S3×C6 C62 C3⋊D4 Dic3 D6 C2×C6 C2×C30 C3×C15 C30 C2×C10 C15 C15 C2×C6 C10 C32 C6 C5 C22 C3 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 4 4 4 8 8 8 8 1 1 1 2 2 2 4 2 4 4 4 8 8 8 8 16

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

 28 0 0 28
,
 1 13 26 29
,
 20 4 16 11
,
 30 18 0 1
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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