direct product, non-abelian, soluble
Aliases: C5×C23.3A4, C10.(C42⋊C3), C2.C42⋊C15, C23.3(C5×A4), (C22×C10).3A4, (C2×C10).SL2(𝔽3), C22.(C5×SL2(𝔽3)), C2.(C5×C42⋊C3), (C5×C2.C42)⋊C3, SmallGroup(480,74)
Series: Derived ►Chief ►Lower central ►Upper central
| C2.C42 — C5×C23.3A4 |
Generators and relations for C5×C23.3A4
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=g3=1, e2=gbg-1=bcd, f2=gcg-1=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef-1=de=ed, df=fd, dg=gd, geg-1=bef, gfg-1=cde >
Smallest permutation representation of C5×C23.3A4
►On 60 points
Generators in S60
(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 21)(2 22)(3 23)(4 24)(5 25)(26 31)(27 32)(28 33)(29 34)(30 35)
(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 56)(7 57)(8 58)(9 59)(10 60)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46 41 51)(37 47 42 52)(38 48 43 53)(39 49 44 54)(40 50 45 55)
(1 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 11)(7 12)(8 13)(9 14)(10 15)(16 56)(17 57)(18 58)(19 59)(20 60)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 41 21)(7 42 22)(8 43 23)(9 44 24)(10 45 25)(11 46 26)(12 47 27)(13 48 28)(14 49 29)(15 50 30)(16 51 31)(17 52 32)(18 53 33)(19 54 34)(20 55 35)
G:=sub<Sym(60)| (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), (1,21)(2,22)(3,23)(4,24)(5,25)(26,31)(27,32)(28,33)(29,34)(30,35), (6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20), (1,21)(2,22)(3,23)(4,24)(5,25)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55), (1,26)(2,27)(3,28)(4,29)(5,30)(6,56)(7,57)(8,58)(9,59)(10,60)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46,41,51)(37,47,42,52)(38,48,43,53)(39,49,44,54)(40,50,45,55), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,11)(7,12)(8,13)(9,14)(10,15)(16,56)(17,57)(18,58)(19,59)(20,60)(36,41)(37,42)(38,43)(39,44)(40,45), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,41,21)(7,42,22)(8,43,23)(9,44,24)(10,45,25)(11,46,26)(12,47,27)(13,48,28)(14,49,29)(15,50,30)(16,51,31)(17,52,32)(18,53,33)(19,54,34)(20,55,35)>;
G:=Group( (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), (1,21)(2,22)(3,23)(4,24)(5,25)(26,31)(27,32)(28,33)(29,34)(30,35), (6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20), (1,21)(2,22)(3,23)(4,24)(5,25)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55), (1,26)(2,27)(3,28)(4,29)(5,30)(6,56)(7,57)(8,58)(9,59)(10,60)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46,41,51)(37,47,42,52)(38,48,43,53)(39,49,44,54)(40,50,45,55), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,11)(7,12)(8,13)(9,14)(10,15)(16,56)(17,57)(18,58)(19,59)(20,60)(36,41)(37,42)(38,43)(39,44)(40,45), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,41,21)(7,42,22)(8,43,23)(9,44,24)(10,45,25)(11,46,26)(12,47,27)(13,48,28)(14,49,29)(15,50,30)(16,51,31)(17,52,32)(18,53,33)(19,54,34)(20,55,35) );
G=PermutationGroup([[(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)], [(1,21),(2,22),(3,23),(4,24),(5,25),(26,31),(27,32),(28,33),(29,34),(30,35)], [(6,56),(7,57),(8,58),(9,59),(10,60),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,56),(7,57),(8,58),(9,59),(10,60),(11,16),(12,17),(13,18),(14,19),(15,20),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,56),(7,57),(8,58),(9,59),(10,60),(21,31),(22,32),(23,33),(24,34),(25,35),(36,46,41,51),(37,47,42,52),(38,48,43,53),(39,49,44,54),(40,50,45,55)], [(1,31,21,26),(2,32,22,27),(3,33,23,28),(4,34,24,29),(5,35,25,30),(6,11),(7,12),(8,13),(9,14),(10,15),(16,56),(17,57),(18,58),(19,59),(20,60),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,56,36),(2,57,37),(3,58,38),(4,59,39),(5,60,40),(6,41,21),(7,42,22),(8,43,23),(9,44,24),(10,45,25),(11,46,26),(12,47,27),(13,48,28),(14,49,29),(15,50,30),(16,51,31),(17,52,32),(18,53,33),(19,54,34),(20,55,35)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 15A | ··· | 15H | 20A | ··· | 20P | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 3 | 3 | 16 | 16 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 16 | 16 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 16 | ··· | 16 | 6 | ··· | 6 | 16 | ··· | 16 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | - | + | + | |||||||||
| image | C1 | C3 | C5 | C15 | SL2(𝔽3) | SL2(𝔽3) | C5×SL2(𝔽3) | A4 | C42⋊C3 | C5×A4 | C5×C42⋊C3 | C23.3A4 | C5×C23.3A4 |
| kernel | C5×C23.3A4 | C5×C2.C42 | C23.3A4 | C2.C42 | C2×C10 | C2×C10 | C22 | C22×C10 | C10 | C23 | C2 | C5 | C1 |
| # reps | 1 | 2 | 4 | 8 | 1 | 2 | 12 | 1 | 4 | 4 | 16 | 1 | 4 |
Matrix representation of C5×C23.3A4 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 48 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 13 | 47 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 48 | 47 | 0 | 0 | 0 |
| 47 | 13 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 9 | 50 |
| 47 | 13 | 0 | 0 | 0 |
| 13 | 14 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 4 | 29 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 47 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 14 | 60 | 35 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,48,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,13,0,0,0,60,47,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[48,47,0,0,0,47,13,0,0,0,0,0,50,0,0,0,0,0,60,9,0,0,0,0,50],[47,13,0,0,0,13,14,0,0,0,0,0,1,0,4,0,0,0,50,29,0,0,0,0,11],[1,47,0,0,0,0,13,0,0,0,0,0,0,14,0,0,0,13,60,0,0,0,0,35,1] >;
C5×C23.3A4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._3A_4
% in TeX
G:=Group("C5xC2^3.3A4"); // GroupNames label
G:=SmallGroup(480,74);
// by ID
G=gap.SmallGroup(480,74);
# by ID
G:=PCGroup([7,-3,-5,-2,2,-2,2,-2,632,268,4623,521,80,12604,17645]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=g^3=1,e^2=g*b*g^-1=b*c*d,f^2=g*c*g^-1=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*e*f,g*f*g^-1=c*d*e>;
// generators/relations
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