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## G = C4×S3×D5order 240 = 24·3·5

### Direct product of C4, S3 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C4×S3×D5
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C4×S3×D5
 Lower central C15 — C4×S3×D5
 Upper central C1 — C4

Generators and relations for C4×S3×D5
G = < a,b,c,d,e | a4=b3=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 448 in 108 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C22 [×7], C5, S3 [×2], S3 [×2], C6, C6 [×2], C2×C4 [×6], C23, D5 [×2], D5 [×2], C10, C10 [×2], Dic3, Dic3, C12, C12, D6, D6 [×5], C2×C6, C15, C22×C4, Dic5, Dic5, C20, C20, D10, D10 [×5], C2×C10, C4×S3, C4×S3 [×3], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C4×D5, C4×D5 [×3], C2×Dic5, C2×C20, C22×D5, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×4], C6×D5, S3×C10, D30, C2×C4×D5, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, C4×S3×D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, C2×S3×D5, C4×S3×D5

Smallest permutation representation of C4×S3×D5
On 60 points
Generators in S60
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)(31 46)(32 47)(33 48)(34 49)(35 50)(36 56)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 53)(44 54)(45 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 18)(2 17)(3 16)(4 20)(5 19)(6 21)(7 25)(8 24)(9 23)(10 22)(11 26)(12 30)(13 29)(14 28)(15 27)(31 46)(32 50)(33 49)(34 48)(35 47)(36 51)(37 55)(38 54)(39 53)(40 52)(41 56)(42 60)(43 59)(44 58)(45 57)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 60A 60B 60C 60D order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 10 10 10 10 10 10 12 12 12 12 15 15 20 20 20 20 20 20 20 20 30 30 60 60 60 60 size 1 1 3 3 5 5 15 15 2 1 1 3 3 5 5 15 15 2 2 2 10 10 2 2 6 6 6 6 2 2 10 10 4 4 2 2 2 2 6 6 6 6 4 4 4 4 4 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D5 D6 D6 D6 D10 D10 D10 C4×S3 C4×D5 S3×D5 C2×S3×D5 C4×S3×D5 kernel C4×S3×D5 D5×Dic3 S3×Dic5 D30.C2 D5×C12 S3×C20 C4×D15 C2×S3×D5 S3×D5 C4×D5 C4×S3 Dic5 C20 D10 Dic3 C12 D6 D5 S3 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 2 2 2 4 8 2 2 4

Matrix representation of C4×S3×D5 in GL4(𝔽61) generated by

 50 0 0 0 0 50 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 1 0 0 0 0 60 60 0 0 1 0
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 1 1
,
 0 1 0 0 60 43 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 18 1 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[60,0,0,0,0,60,0,0,0,0,60,1,0,0,0,1],[0,60,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[60,18,0,0,0,1,0,0,0,0,60,0,0,0,0,60] >;

C4×S3×D5 in GAP, Magma, Sage, TeX

C_4\times S_3\times D_5
% in TeX

G:=Group("C4xS3xD5");
// GroupNames label

G:=SmallGroup(240,135);
// by ID

G=gap.SmallGroup(240,135);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^5=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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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