Copied to
clipboard

## G = C5×A4⋊C4order 240 = 24·3·5

### Direct product of C5 and A4⋊C4

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

 Derived series C1 — C22 — A4 — C5×A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — C10×A4 — C5×A4⋊C4
 Lower central A4 — C5×A4⋊C4
 Upper central C1 — C10

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  Dic52S4  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

Export

׿
×
𝔽