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G = C22xS3xF5order 480 = 25·3·5

Direct product of C22, S3 and F5

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

Aliases: C22xS3xF5, C3:F5:C23, C15:(C23xC4), C30:(C22xC4), (C3xF5):C23, C3:1(C23xF5), D30:7(C2xC4), D15:(C22xC4), (S3xD5).C23, (C3xD5).C24, C6:1(C22xF5), D10:14(C4xS3), (C6xF5):3C22, (C22xD15):6C4, D5.1(S3xC23), (C6xD5).36C23, (C22xD5).78D6, D10.39(C22xS3), C10:(S3xC2xC4), D5:(S3xC2xC4), C5:(S3xC22xC4), (S3xD5):(C2xC4), (C2xS3xD5):6C4, (C2xC6xF5):3C2, (S3xC2xC10):5C4, (C2xC6):6(C2xF5), (C2xC30):4(C2xC4), (C2xC10):9(C4xS3), (C5xS3):(C22xC4), (C3xD5):(C22xC4), (S3xC10):7(C2xC4), (C22xC3:F5):3C2, (C2xC3:F5):3C22, (C6xD5):14(C2xC4), (C22xS3xD5).6C2, (C2xS3xD5).20C22, (D5xC2xC6).73C22, SmallGroup(480,1197)

Series: Derived Chief Lower central Upper central

C1C15 — C22xS3xF5
C1C5C15C3xD5C3xF5S3xF5C2xS3xF5 — C22xS3xF5
C15 — C22xS3xF5
C1C22

Generators and relations for C22xS3xF5
 G = < a,b,c,d,e,f | a2=b2=c3=d2=e5=f4=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-1=e3 >

Subgroups: 2292 in 472 conjugacy classes, 166 normal (20 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, S3, C6, C6, C2xC4, C23, D5, D5, D5, C10, C10, Dic3, C12, D6, D6, C2xC6, C2xC6, C15, C22xC4, C24, F5, F5, D10, D10, C2xC10, C2xC10, C4xS3, C2xDic3, C2xC12, C22xS3, C22xS3, C22xC6, C5xS3, C3xD5, C3xD5, D15, C30, C23xC4, C2xF5, C2xF5, C22xD5, C22xD5, C22xC10, S3xC2xC4, C22xDic3, C22xC12, S3xC23, C3xF5, C3:F5, S3xD5, C6xD5, S3xC10, D30, C2xC30, C22xF5, C22xF5, C23xD5, S3xC22xC4, S3xF5, C6xF5, C2xC3:F5, C2xS3xD5, D5xC2xC6, S3xC2xC10, C22xD15, C23xF5, C2xS3xF5, C2xC6xF5, C22xC3:F5, C22xS3xD5, C22xS3xF5
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C24, F5, C4xS3, C22xS3, C23xC4, C2xF5, S3xC2xC4, S3xC23, C22xF5, S3xC22xC4, S3xF5, C23xF5, C2xS3xF5, C22xS3xF5

Smallest permutation representation of C22xS3xF5
On 60 points
Generators in S60
(1 31)(2 32)(3 33)(4 34)(5 35)(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 16)(2 17)(3 18)(4 19)(5 20)(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 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 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 46)(2 47)(3 48)(4 49)(5 50)(6 56)(7 57)(8 58)(9 59)(10 60)(11 51)(12 52)(13 53)(14 54)(15 55)(16 31)(17 32)(18 33)(19 34)(20 35)(21 41)(22 42)(23 43)(24 44)(25 45)(26 36)(27 37)(28 38)(29 39)(30 40)
(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 46)(2 48 5 49)(3 50 4 47)(6 51)(7 53 10 54)(8 55 9 52)(11 56)(12 58 15 59)(13 60 14 57)(16 31)(17 33 20 34)(18 35 19 32)(21 36)(22 38 25 39)(23 40 24 37)(26 41)(27 43 30 44)(28 45 29 42)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A···4H4I···4P 5 6A6B6C6D6E6F6G10A10B10C10D10E10F10G12A···12H 15 30A30B30C
order122222222222222234···44···4566666661010101010101012···1215303030
size1111333355551515151525···515···154222101010104441212121210···108888

60 irreducible representations

dim111111112222244488
type+++++++++++++
imageC1C2C2C2C2C4C4C4S3D6D6C4xS3C4xS3F5C2xF5C2xF5S3xF5C2xS3xF5
kernelC22xS3xF5C2xS3xF5C2xC6xF5C22xC3:F5C22xS3xD5C2xS3xD5S3xC2xC10C22xD15C22xF5C2xF5C22xD5D10C2xC10C22xS3D6C2xC6C22C2
# reps11211112221616216113

Matrix representation of C22xS3xF5 in GL8(F61)

10000000
01000000
006000000
000600000
00001000
00000100
00000010
00000001
,
600000000
060000000
00100000
00010000
00001000
00000100
00000010
00000001
,
060000000
160000000
0060600000
00100000
00001000
00000100
00000010
00000001
,
601000000
01000000
006000000
00110000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000000060
000010060
000001060
000000160
,
110000000
011000000
005000000
000500000
00000010
00001000
00000001
00000100

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C22xS3xF5 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times F_5
% in TeX

G:=Group("C2^2xS3xF5");
// GroupNames label

G:=SmallGroup(480,1197);
// by ID

G=gap.SmallGroup(480,1197);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,1356,9414,1210]);
// Polycyclic

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

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