direct product, non-abelian, soluble, monomial
Aliases: C5×A4⋊C4, A4⋊C20, C10.6S4, (C5×A4)⋊5C4, (C2×A4).C10, C2.1(C5×S4), C23.(C5×S3), C22⋊(C5×Dic3), (C10×A4).3C2, (C2×C10)⋊2Dic3, (C22×C10).1S3, SmallGroup(240,104)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C5×A4⋊C4 |
Generators and relations for C5×A4⋊C4
G = < a,b,c,d,e | a5=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >
Smallest permutation representation of C5×A4⋊C4
►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 8)(2 9)(3 10)(4 6)(5 7)(11 48)(12 49)(13 50)(14 46)(15 47)(16 37)(17 38)(18 39)(19 40)(20 36)(21 60)(22 56)(23 57)(24 58)(25 59)(26 52)(27 53)(28 54)(29 55)(30 51)(31 41)(32 42)(33 43)(34 44)(35 45)
(1 58)(2 59)(3 60)(4 56)(5 57)(6 22)(7 23)(8 24)(9 25)(10 21)(11 26)(12 27)(13 28)(14 29)(15 30)(16 41)(17 42)(18 43)(19 44)(20 45)(31 37)(32 38)(33 39)(34 40)(35 36)(46 55)(47 51)(48 52)(49 53)(50 54)
(1 35 11)(2 31 12)(3 32 13)(4 33 14)(5 34 15)(6 39 55)(7 40 51)(8 36 52)(9 37 53)(10 38 54)(16 49 59)(17 50 60)(18 46 56)(19 47 57)(20 48 58)(21 42 28)(22 43 29)(23 44 30)(24 45 26)(25 41 27)
(1 24 58 8)(2 25 59 9)(3 21 60 10)(4 22 56 6)(5 23 57 7)(11 45 48 36)(12 41 49 37)(13 42 50 38)(14 43 46 39)(15 44 47 40)(16 53 31 27)(17 54 32 28)(18 55 33 29)(19 51 34 30)(20 52 35 26)
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), (1,8)(2,9)(3,10)(4,6)(5,7)(11,48)(12,49)(13,50)(14,46)(15,47)(16,37)(17,38)(18,39)(19,40)(20,36)(21,60)(22,56)(23,57)(24,58)(25,59)(26,52)(27,53)(28,54)(29,55)(30,51)(31,41)(32,42)(33,43)(34,44)(35,45), (1,58)(2,59)(3,60)(4,56)(5,57)(6,22)(7,23)(8,24)(9,25)(10,21)(11,26)(12,27)(13,28)(14,29)(15,30)(16,41)(17,42)(18,43)(19,44)(20,45)(31,37)(32,38)(33,39)(34,40)(35,36)(46,55)(47,51)(48,52)(49,53)(50,54), (1,35,11)(2,31,12)(3,32,13)(4,33,14)(5,34,15)(6,39,55)(7,40,51)(8,36,52)(9,37,53)(10,38,54)(16,49,59)(17,50,60)(18,46,56)(19,47,57)(20,48,58)(21,42,28)(22,43,29)(23,44,30)(24,45,26)(25,41,27), (1,24,58,8)(2,25,59,9)(3,21,60,10)(4,22,56,6)(5,23,57,7)(11,45,48,36)(12,41,49,37)(13,42,50,38)(14,43,46,39)(15,44,47,40)(16,53,31,27)(17,54,32,28)(18,55,33,29)(19,51,34,30)(20,52,35,26)>;
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), (1,8)(2,9)(3,10)(4,6)(5,7)(11,48)(12,49)(13,50)(14,46)(15,47)(16,37)(17,38)(18,39)(19,40)(20,36)(21,60)(22,56)(23,57)(24,58)(25,59)(26,52)(27,53)(28,54)(29,55)(30,51)(31,41)(32,42)(33,43)(34,44)(35,45), (1,58)(2,59)(3,60)(4,56)(5,57)(6,22)(7,23)(8,24)(9,25)(10,21)(11,26)(12,27)(13,28)(14,29)(15,30)(16,41)(17,42)(18,43)(19,44)(20,45)(31,37)(32,38)(33,39)(34,40)(35,36)(46,55)(47,51)(48,52)(49,53)(50,54), (1,35,11)(2,31,12)(3,32,13)(4,33,14)(5,34,15)(6,39,55)(7,40,51)(8,36,52)(9,37,53)(10,38,54)(16,49,59)(17,50,60)(18,46,56)(19,47,57)(20,48,58)(21,42,28)(22,43,29)(23,44,30)(24,45,26)(25,41,27), (1,24,58,8)(2,25,59,9)(3,21,60,10)(4,22,56,6)(5,23,57,7)(11,45,48,36)(12,41,49,37)(13,42,50,38)(14,43,46,39)(15,44,47,40)(16,53,31,27)(17,54,32,28)(18,55,33,29)(19,51,34,30)(20,52,35,26) );
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)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,48),(12,49),(13,50),(14,46),(15,47),(16,37),(17,38),(18,39),(19,40),(20,36),(21,60),(22,56),(23,57),(24,58),(25,59),(26,52),(27,53),(28,54),(29,55),(30,51),(31,41),(32,42),(33,43),(34,44),(35,45)], [(1,58),(2,59),(3,60),(4,56),(5,57),(6,22),(7,23),(8,24),(9,25),(10,21),(11,26),(12,27),(13,28),(14,29),(15,30),(16,41),(17,42),(18,43),(19,44),(20,45),(31,37),(32,38),(33,39),(34,40),(35,36),(46,55),(47,51),(48,52),(49,53),(50,54)], [(1,35,11),(2,31,12),(3,32,13),(4,33,14),(5,34,15),(6,39,55),(7,40,51),(8,36,52),(9,37,53),(10,38,54),(16,49,59),(17,50,60),(18,46,56),(19,47,57),(20,48,58),(21,42,28),(22,43,29),(23,44,30),(24,45,26),(25,41,27)], [(1,24,58,8),(2,25,59,9),(3,21,60,10),(4,22,56,6),(5,23,57,7),(11,45,48,36),(12,41,49,37),(13,42,50,38),(14,43,46,39),(15,44,47,40),(16,53,31,27),(17,54,32,28),(18,55,33,29),(19,51,34,30),(20,52,35,26)]])
C5×A4⋊C4 is a maximal subgroup of
A4⋊Dic10 Dic5⋊2S4 A4⋊D20 C20×S4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | 30B | 30C | 30D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | 30 | 30 | 30 |
| size | 1 | 1 | 3 | 3 | 8 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | S3 | Dic3 | C5×S3 | C5×Dic3 | S4 | A4⋊C4 | C5×S4 | C5×A4⋊C4 |
| kernel | C5×A4⋊C4 | C10×A4 | C5×A4 | A4⋊C4 | C2×A4 | A4 | C22×C10 | C2×C10 | C23 | C22 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 |
Matrix representation of C5×A4⋊C4 ►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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 60 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 11 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 60 | 0 |
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],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[60,1,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,11,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,60,0] >;
C5×A4⋊C4 in GAP, Magma, Sage, TeX
C_5\times A_4\rtimes C_4
% in TeX
G:=Group("C5xA4:C4"); // GroupNames label
G:=SmallGroup(240,104);
// by ID
G=gap.SmallGroup(240,104);
# by ID
G:=PCGroup([6,-2,-5,-2,-3,-2,2,60,963,3604,202,2165,347]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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