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## G = C3×C15⋊7D4order 360 = 23·32·5

### Direct product of C3 and C15⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C3×C15⋊7D4
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C6×D15 — C3×C15⋊7D4
 Lower central C15 — C30 — C3×C15⋊7D4
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×C157D4
G = < a,b,c,d | a3=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 300 in 74 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C15, C3×S3, C3×C6, C3×C6, Dic5, D10, C2×C10, C3⋊D4, C3×D4, C3×D5, D15, C30, C30, C3×Dic3, S3×C6, C62, C5⋊D4, C3×C15, C3×Dic5, Dic15, C6×D5, D30, C2×C30, C2×C30, C3×C3⋊D4, C3×D15, C3×C30, C3×C30, C3×C5⋊D4, C157D4, C3×Dic15, C6×D15, C6×C30, C3×C157D4
Quotients: C1, C2, C3, C22, S3, C6, D4, D5, D6, C2×C6, C3×S3, D10, C3⋊D4, C3×D4, C3×D5, D15, S3×C6, C5⋊D4, C6×D5, D30, C3×C3⋊D4, C3×D15, C3×C5⋊D4, C157D4, C6×D15, C3×C157D4

Smallest permutation representation of C3×C157D4
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 54 27 32)(2 53 28 31)(3 52 29 45)(4 51 30 44)(5 50 16 43)(6 49 17 42)(7 48 18 41)(8 47 19 40)(9 46 20 39)(10 60 21 38)(11 59 22 37)(12 58 23 36)(13 57 24 35)(14 56 25 34)(15 55 26 33)
(1 32)(2 31)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 50)(17 49)(18 48)(19 47)(20 46)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 5A 5B 6A 6B 6C ··· 6M 6N 6O 10A ··· 10F 12A 12B 15A ··· 15P 30A ··· 30AV order 1 2 2 2 3 3 3 3 3 4 5 5 6 6 6 ··· 6 6 6 10 ··· 10 12 12 15 ··· 15 30 ··· 30 size 1 1 2 30 1 1 2 2 2 30 2 2 1 1 2 ··· 2 30 30 2 ··· 2 30 30 2 ··· 2 2 ··· 2

99 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 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D5 D6 C3×S3 D10 C3⋊D4 C3×D4 C3×D5 D15 S3×C6 C5⋊D4 C6×D5 D30 C3×C3⋊D4 C3×D15 C3×C5⋊D4 C15⋊7D4 C6×D15 C3×C15⋊7D4 kernel C3×C15⋊7D4 C3×Dic15 C6×D15 C6×C30 C15⋊7D4 Dic15 D30 C2×C30 C2×C30 C3×C15 C62 C30 C2×C10 C3×C6 C15 C15 C2×C6 C2×C6 C10 C32 C6 C6 C5 C22 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 2 1 2 2 2 2 4 4 2 4 4 4 4 8 8 8 8 16

Matrix representation of C3×C157D4 in GL2(𝔽31) generated by

 25 0 0 25
,
 18 3 13 16
,
 0 6 5 0
,
 26 29 12 5
G:=sub<GL(2,GF(31))| [25,0,0,25],[18,13,3,16],[0,5,6,0],[26,12,29,5] >;

C3×C157D4 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes_7D_4
% in TeX

G:=Group("C3xC15:7D4");
// GroupNames label

G:=SmallGroup(360,104);
// by ID

G=gap.SmallGroup(360,104);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,1444,10373]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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