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G = C2×S3×D15order 360 = 23·32·5

Direct product of C2, S3 and D15

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

Aliases: C2×S3×D15, C303D6, C61D30, C101S32, C61(S3×D5), (C5×S3)⋊2D6, (S3×C6)⋊5D5, (C3×C6)⋊1D10, (S3×C30)⋊6C2, (S3×C10)⋊3S3, (C3×S3)⋊2D10, (C3×C15)⋊4C23, (C6×D15)⋊10C2, (C3×C30)⋊3C22, C154(C22×S3), C3⋊D153C22, C31(C22×D15), C322(C22×D5), (S3×C15)⋊2C22, (C3×D15)⋊3C22, C52(C2×S32), C33(C2×S3×D5), (C2×C3⋊D15)⋊8C2, SmallGroup(360,154)

Series: Derived Chief Lower central Upper central

C1C3×C15 — C2×S3×D15
C1C5C15C3×C15C3×D15S3×D15 — C2×S3×D15
C3×C15 — C2×S3×D15
C1C2

Generators and relations for C2×S3×D15
 G = < a,b,c,d,e | a2=b3=c2=d15=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: 1092 in 138 conjugacy classes, 42 normal (24 characteristic)
C1, C2, C2, C3, C3, C22, C5, S3, S3, C6, C6, C23, C32, D5, C10, C10, D6, D6, C2×C6, C15, C15, C3×S3, C3×S3, C3⋊S3, C3×C6, D10, C2×C10, C22×S3, C5×S3, C3×D5, D15, D15, C30, C30, S32, S3×C6, S3×C6, C2×C3⋊S3, C22×D5, C3×C15, S3×D5, C6×D5, S3×C10, D30, D30, C2×C30, C2×S32, S3×C15, C3×D15, C3⋊D15, C3×C30, C2×S3×D5, C22×D15, S3×D15, S3×C30, C6×D15, C2×C3⋊D15, C2×S3×D15
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22×S3, D15, S32, C22×D5, S3×D5, D30, C2×S32, C2×S3×D5, C22×D15, S3×D15, C2×S3×D15

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F15A15B15C15D15E···15J30A30B30C30D30E···30J30K···30R
order122222223335566666661010101010101515151515···153030303030···3030···30
size1133151545452242222466303022666622224···422224···46···6

54 irreducible representations

dim1111122222222222444444
type++++++++++++++++++++++
imageC1C2C2C2C2S3S3D5D6D6D6D10D10D15D30D30S32S3×D5C2×S32C2×S3×D5S3×D15C2×S3×D15
kernelC2×S3×D15S3×D15S3×C30C6×D15C2×C3⋊D15S3×C10D30S3×C6C5×S3D15C30C3×S3C3×C6D6S3C6C10C6C5C3C2C1
# reps1411111222242484121244

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

100000
010000
0030000
0003000
000010
000001
,
100000
010000
001000
000100
0000301
0000300
,
3000000
0300000
0030000
0003000
000001
000010
,
13180000
13300000
0003000
0013000
000010
000001
,
0300000
3000000
0030100
000100
000010
000001

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,13,0,0,0,0,18,30,0,0,0,0,0,0,0,1,0,0,0,0,30,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,30,0,0,0,0,30,0,0,0,0,0,0,0,30,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×S3×D15 in GAP, Magma, Sage, TeX

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

G:=Group("C2xS3xD15");
// GroupNames label

G:=SmallGroup(360,154);
// by ID

G=gap.SmallGroup(360,154);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,201,1444,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^15=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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