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## G = C2×D15⋊S3order 360 = 23·32·5

### Direct product of C2 and D15⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C2×D15⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C3×D15 — D15⋊S3 — C2×D15⋊S3
 Lower central C3×C15 — C2×D15⋊S3
 Upper central C1 — C2

Generators and relations for C2×D15⋊S3
G = < a,b,c,d,e | a2=b15=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b11, cd=dc, ece=b10c, ede=d-1 >

Subgroups: 924 in 138 conjugacy classes, 39 normal (11 characteristic)
C1, C2, C2, C3, C3, C22, C5, S3, C6, C6, C23, C32, D5, C10, C10, D6, C2×C6, C15, C15, C3×S3, C3⋊S3, C3×C6, D10, C2×C10, C22×S3, C5×S3, C3×D5, D15, C30, C30, S32, S3×C6, C2×C3⋊S3, C22×D5, C3×C15, S3×D5, C6×D5, S3×C10, D30, C2×S32, C3×D15, C5×C3⋊S3, C3×C30, C2×S3×D5, D15⋊S3, C6×D15, C10×C3⋊S3, C2×D15⋊S3
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22×S3, S32, C22×D5, S3×D5, C2×S32, C2×S3×D5, D15⋊S3, C2×D15⋊S3

Smallest permutation representation of C2×D15⋊S3
On 60 points
Generators in S60
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(31 58)(32 59)(33 60)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)
(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)
(2 12)(3 8)(5 15)(6 11)(9 14)(16 21)(18 28)(19 24)(22 27)(25 30)(31 41)(32 37)(34 44)(35 40)(38 43)(46 56)(47 52)(49 59)(50 55)(53 58)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 15A ··· 15H 30A ··· 30H order 1 2 2 2 2 2 2 2 3 3 3 5 5 6 6 6 6 6 6 6 10 10 10 10 10 10 15 ··· 15 30 ··· 30 size 1 1 9 9 15 15 15 15 2 2 4 2 2 2 2 4 30 30 30 30 2 2 18 18 18 18 4 ··· 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D5 D6 D6 D10 D10 S32 S3×D5 C2×S32 C2×S3×D5 D15⋊S3 C2×D15⋊S3 kernel C2×D15⋊S3 D15⋊S3 C6×D15 C10×C3⋊S3 D30 C2×C3⋊S3 D15 C30 C3⋊S3 C3×C6 C10 C6 C5 C3 C2 C1 # reps 1 4 2 1 2 2 4 2 4 2 1 4 1 4 4 4

Matrix representation of C2×D15⋊S3 in GL6(𝔽31)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 17 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 1 0 0 0 0 30 0
,
 1 1 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 30 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 1 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,17,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[1,0,0,0,0,0,1,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,30,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D15⋊S3 in GAP, Magma, Sage, TeX

C_2\times D_{15}\rtimes S_3
% in TeX

G:=Group("C2xD15:S3");
// GroupNames label

G:=SmallGroup(360,155);
// by ID

G=gap.SmallGroup(360,155);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,387,201,730,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^15=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^11,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
// generators/relations

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