direct product, metabelian, soluble, monomial, A-group
Aliases: C5×C42⋊C3, C42⋊C15, (C4×C20)⋊C3, (C2×C10).A4, C22.(C5×A4), SmallGroup(240,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C5×C42⋊C3 |
Generators and relations for C5×C42⋊C3
G = < a,b,c,d | a5=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >
Smallest permutation representation of C5×C42⋊C3
►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 26 21 31)(2 27 22 32)(3 28 23 33)(4 29 24 34)(5 30 25 35)(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)(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 16 56 11)(7 17 57 12)(8 18 58 13)(9 19 59 14)(10 20 60 15)
(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), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(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)(6,16,56,11)(7,17,57,12)(8,18,58,13)(9,19,59,14)(10,20,60,15), (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), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(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)(6,16,56,11)(7,17,57,12)(8,18,58,13)(9,19,59,14)(10,20,60,15), (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)], [(1,26,21,31),(2,27,22,32),(3,28,23,33),(4,29,24,34),(5,30,25,35),(6,56),(7,57),(8,58),(9,59),(10,60),(11,16),(12,17),(13,18),(14,19),(15,20),(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),(6,16,56,11),(7,17,57,12),(8,18,58,13),(9,19,59,14),(10,20,60,15)], [(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)]])
C5×C42⋊C3 is a maximal subgroup of
C42⋊D15 C20⋊4D4⋊C3 (C4×C20)⋊C6
40 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 20A | ··· | 20P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 20 | ··· | 20 |
| size | 1 | 3 | 16 | 16 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 16 | ··· | 16 | 3 | ··· | 3 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||
| image | C1 | C3 | C5 | C15 | A4 | C42⋊C3 | C5×A4 | C5×C42⋊C3 |
| kernel | C5×C42⋊C3 | C4×C20 | C42⋊C3 | C42 | C2×C10 | C5 | C22 | C1 |
| # reps | 1 | 2 | 4 | 8 | 1 | 4 | 4 | 16 |
Matrix representation of C5×C42⋊C3 ►in GL3(𝔽61) generated by
| 58 | 0 | 0 |
| 0 | 58 | 0 |
| 0 | 0 | 58 |
| 50 | 0 | 0 |
| 0 | 50 | 0 |
| 0 | 0 | 60 |
| 11 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 50 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(61))| [58,0,0,0,58,0,0,0,58],[50,0,0,0,50,0,0,0,60],[11,0,0,0,1,0,0,0,50],[0,0,1,1,0,0,0,1,0] >;
C5×C42⋊C3 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes C_3
% in TeX
G:=Group("C5xC4^2:C3"); // GroupNames label
G:=SmallGroup(240,32);
// by ID
G=gap.SmallGroup(240,32);
# by ID
G:=PCGroup([6,-3,-5,-2,2,-2,2,542,230,3963,69,3604,6485]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations
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