Copied to
clipboard

G = C22×S3×D5order 240 = 24·3·5

Direct product of C22, S3 and D5

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

Aliases: C22×S3×D5, C15⋊C24, C30⋊C23, D15⋊C23, D3011C22, (C2×C6)⋊6D10, (C2×C10)⋊9D6, (C5×S3)⋊C23, C51(S3×C23), (C3×D5)⋊C23, C31(C23×D5), C61(C22×D5), (C2×C30)⋊5C22, C101(C22×S3), (C6×D5)⋊8C22, (S3×C10)⋊8C22, (C22×D15)⋊7C2, (D5×C2×C6)⋊4C2, (S3×C2×C10)⋊4C2, SmallGroup(240,202)

Series: Derived Chief Lower central Upper central

C1C15 — C22×S3×D5
C1C5C15C3×D5S3×D5C2×S3×D5 — C22×S3×D5
C15 — C22×S3×D5
C1C22

Generators and relations for C22×S3×D5
 G = < a,b,c,d,e,f | a2=b2=c3=d2=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1200 in 268 conjugacy classes, 104 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C3, C22, C22 [×34], C5, S3 [×4], S3 [×4], C6 [×3], C6 [×4], C23 [×15], D5 [×4], D5 [×4], C10 [×3], C10 [×4], D6 [×6], D6 [×22], C2×C6, C2×C6 [×6], C15, C24, D10 [×6], D10 [×22], C2×C10, C2×C10 [×6], C22×S3, C22×S3 [×13], C22×C6, C5×S3 [×4], C3×D5 [×4], D15 [×4], C30 [×3], C22×D5, C22×D5 [×13], C22×C10, S3×C23, S3×D5 [×16], C6×D5 [×6], S3×C10 [×6], D30 [×6], C2×C30, C23×D5, C2×S3×D5 [×12], D5×C2×C6, S3×C2×C10, C22×D15, C22×S3×D5
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], C22×D5 [×7], S3×C23, S3×D5, C23×D5, C2×S3×D5 [×3], C22×S3×D5

Smallest permutation representation of C22×S3×D5
On 60 points
Generators in S60
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)
(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 34)(2 35)(3 31)(4 32)(5 33)(6 41)(7 42)(8 43)(9 44)(10 45)(11 36)(12 37)(13 38)(14 39)(15 40)(16 46)(17 47)(18 48)(19 49)(20 50)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 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 33)(2 32)(3 31)(4 35)(5 34)(6 36)(7 40)(8 39)(9 38)(10 37)(11 41)(12 45)(13 44)(14 43)(15 42)(16 46)(17 50)(18 49)(19 48)(20 47)(21 51)(22 55)(23 54)(24 53)(25 52)(26 56)(27 60)(28 59)(29 58)(30 57)

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

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

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

C22×S3×D5 is a maximal subgroup of
D30.27D4  D64D20  D304D4  D305D4  D308D4
C22×S3×D5 is a maximal quotient of
D20.38D6  D20.39D6  C30.C24  D2024D6  D2025D6  D2026D6  D2029D6  C15⋊2- 1+4  D30.C23  D2013D6  D2014D6  D1214D10  D20.29D6  C30.33C24  D12.29D10  D2016D6  D2017D6  C15⋊2+ 1+4

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 5A5B6A6B6C6D6E6F6G10A···10F10G···10N15A15B30A···30F
order1222222222222222355666666610···1010···10151530···30
size11113333555515151515222222101010102···26···6444···4

48 irreducible representations

dim1111122222244
type+++++++++++++
imageC1C2C2C2C2S3D5D6D6D10D10S3×D5C2×S3×D5
kernelC22×S3×D5C2×S3×D5D5×C2×C6S3×C2×C10C22×D15C22×D5C22×S3D10C2×C10D6C2×C6C22C2
# reps112111126112226

Matrix representation of C22×S3×D5 in GL4(𝔽31) generated by

30000
03000
0010
0001
,
30000
03000
00300
00030
,
1000
0100
002926
00131
,
30000
03000
0010
001830
,
30100
171300
0010
0001
,
1000
143000
0010
0001
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,1,0,0,0,0,29,13,0,0,26,1],[30,0,0,0,0,30,0,0,0,0,1,18,0,0,0,30],[30,17,0,0,1,13,0,0,0,0,1,0,0,0,0,1],[1,14,0,0,0,30,0,0,0,0,1,0,0,0,0,1] >;

C22×S3×D5 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times D_5
% in TeX

G:=Group("C2^2xS3xD5");
// GroupNames label

G:=SmallGroup(240,202);
// by ID

G=gap.SmallGroup(240,202);
# by ID

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

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

׿
×
𝔽