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G = C4×S3×D5order 240 = 24·3·5

Direct product of C4, S3 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×S3×D5, C205D6, C125D10, C605C22, Dic55D6, D6.9D10, Dic35D10, D10.17D6, C30.11C23, Dic155C22, D30.12C22, (S3×C20)⋊5C2, D152(C2×C4), (C4×D15)⋊9C2, (D5×C12)⋊5C2, C152(C22×C4), D30.C26C2, (D5×Dic3)⋊6C2, (S3×Dic5)⋊6C2, C6.11(C22×D5), (S3×C10).9C22, C10.11(C22×S3), (C5×Dic3)⋊3C22, (C3×Dic5)⋊3C22, (C6×D5).13C22, C52(S3×C2×C4), C31(C2×C4×D5), C2.1(C2×S3×D5), (C2×S3×D5).3C2, (C5×S3)⋊2(C2×C4), (C3×D5)⋊2(C2×C4), SmallGroup(240,135)

Series: Derived Chief Lower central Upper central

C1C15 — C4×S3×D5
C1C5C15C30C6×D5C2×S3×D5 — C4×S3×D5
C15 — C4×S3×D5
C1C4

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)])

C4×S3×D5 is a maximal subgroup of
C40⋊D6  C4⋊F53S3  (C4×S3)⋊F5  D15⋊M4(2)  C5⋊C8⋊D6  D2024D6  D30.C23  D2016D6
C4×S3×D5 is a maximal quotient of
C40⋊D6  C40.54D6  C40.34D6  C40.55D6  C40.35D6  Dic55Dic6  Dic35Dic10  Dic155Q8  (D5×Dic3)⋊C4  D10.19(C4×S3)  Dic34D20  Dic1513D4  D6.(C4×D5)  (S3×Dic5)⋊C4  D30.C2⋊C4  D30.23(C2×C4)  D30.Q8  Dic54D12  Dic1514D4  D6⋊(C4×D5)  C1517(C4×D4)  Dic159D4  C1520(C4×D4)  C1522(C4×D4)  D30.27D4

48 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A10B10C10D10E10F12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A30B60A60B60C60D
order12222222344444444556661010101010101212121215152020202020202020303060606060
size11335515152113355151522210102266662210104422226666444444

48 irreducible representations

dim1111111112222222222444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4S3D5D6D6D6D10D10D10C4×S3C4×D5S3×D5C2×S3×D5C4×S3×D5
kernelC4×S3×D5D5×Dic3S3×Dic5D30.C2D5×C12S3×C20C4×D15C2×S3×D5S3×D5C4×D5C4×S3Dic5C20D10Dic3C12D6D5S3C4C2C1
# reps1111111181211122248224

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

50000
05000
00110
00011
,
1000
0100
006060
0010
,
60000
06000
00600
0011
,
0100
604300
0010
0001
,
60000
18100
00600
00060
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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