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G = C22×S3×F5  order 480 = 25·3·5

Direct product of C22, S3 and F5

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

Aliases: C22×S3×F5, C3⋊F5⋊C23, C15⋊(C23×C4), C30⋊(C22×C4), (C3×F5)⋊C23, C3⋊1(C23×F5), D30⋊7(C2×C4), D15⋊(C22×C4), (C3×D5).C24, (S3×D5).C23, C6⋊1(C22×F5), D10⋊14(C4×S3), (C6×F5)⋊3C22, (C22×D15)⋊6C4, D5.1(S3×C23), (C22×D5).78D6, (C6×D5).36C23, D10.39(C22×S3), C10⋊(S3×C2×C4), D5⋊(S3×C2×C4), C5⋊(S3×C22×C4), (S3×D5)⋊(C2×C4), (C2×S3×D5)⋊6C4, (C2×C6×F5)⋊3C2, (S3×C2×C10)⋊5C4, (C2×C6)⋊6(C2×F5), (C2×C30)⋊4(C2×C4), (C2×C10)⋊9(C4×S3), (C5×S3)⋊(C22×C4), (C3×D5)⋊(C22×C4), (S3×C10)⋊7(C2×C4), (C22×C3⋊F5)⋊3C2, (C2×C3⋊F5)⋊3C22, (C6×D5)⋊14(C2×C4), (C22×S3×D5).6C2, (C2×S3×D5).20C22, (D5×C2×C6).73C22, SmallGroup(480,1197)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C22×S3×F5
Chief series
C1 — C5 — C15 — C3×D5 — C3×F5 — S3×F5 — C2×S3×F5 — C22×S3×F5
Lower central
C15 — C22×S3×F5

Subgroups: 2292 in 472 conjugacy classes, 166 normal (20 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×8], C22, C22 [×34], C5, S3 [×4], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×28], C23 [×15], D5, D5 [×3], D5 [×4], C10 [×3], C10 [×4], Dic3 [×4], C12 [×4], D6 [×6], D6 [×22], C2×C6, C2×C6 [×6], C15, C22×C4 [×14], C24, F5 [×4], F5 [×4], D10 [×6], D10 [×22], C2×C10, C2×C10 [×6], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×S3 [×13], C22×C6, C5×S3 [×4], C3×D5, C3×D5 [×3], D15 [×4], C30 [×3], C23×C4, C2×F5 [×6], C2×F5 [×22], C22×D5, C22×D5 [×13], C22×C10, S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, C3×F5 [×4], C3⋊F5 [×4], S3×D5 [×16], C6×D5 [×6], S3×C10 [×6], D30 [×6], C2×C30, C22×F5, C22×F5 [×13], C23×D5, S3×C22×C4, S3×F5 [×16], C6×F5 [×6], C2×C3⋊F5 [×6], C2×S3×D5 [×12], D5×C2×C6, S3×C2×C10, C22×D15, C23×F5, C2×S3×F5 [×12], C2×C6×F5, C22×C3⋊F5, C22×S3×D5, C22×S3×F5

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, F5, C4×S3 [×4], C22×S3 [×7], C23×C4, C2×F5 [×7], S3×C2×C4 [×6], S3×C23, C22×F5 [×7], S3×C22×C4, S3×F5, C23×F5, C2×S3×F5 [×3], C22×S3×F5

Generators and relations
 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 >

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ 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] >;
 

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+++++++++++++
imageC1C2C2C2C2C4C4C4S3D6D6C4×S3C4×S3F5C2×F5C2×F5S3×F5C2×S3×F5
kernelC22×S3×F5C2×S3×F5C2×C6×F5C22×C3⋊F5C22×S3×D5C2×S3×D5S3×C2×C10C22×D15C22×F5C2×F5C22×D5D10C2×C10C22×S3D6C2×C6C22C2
# reps11211112221616216113

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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