Copied to
clipboard

## G = C4×C5⋊S4order 480 = 25·3·5

### Direct product of C4 and C5⋊S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C4×C5⋊S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — C2×C5⋊S4 — C4×C5⋊S4
 Lower central C5×A4 — C4×C5⋊S4
 Upper central C1 — C4

Generators and relations for C4×C5⋊S4
G = < a,b,c,d,e,f | a4=b5=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 864 in 112 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, A4, D6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×S3, S4, C2×A4, D15, C30, C4×D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, A4⋊C4, C4×A4, C2×S4, Dic15, C60, C5×A4, D30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×S4, C4×D15, C5⋊S4, C10×A4, C4×C5⋊D4, A4⋊Dic5, A4×C20, C2×C5⋊S4, C4×C5⋊S4
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, D6, D10, C4×S3, S4, D15, C4×D5, C2×S4, D30, C4×S4, C4×D15, C5⋊S4, C2×C5⋊S4, C4×C5⋊S4

Smallest permutation representation of C4×C5⋊S4
On 60 points
Generators in S60
(1 43 13 38)(2 44 14 39)(3 45 15 40)(4 41 11 36)(5 42 12 37)(6 31 56 26)(7 32 57 27)(8 33 58 28)(9 34 59 29)(10 35 60 30)(16 51 21 46)(17 52 22 47)(18 53 23 48)(19 54 24 49)(20 55 25 50)
(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 13)(2 14)(3 15)(4 11)(5 12)(6 56)(7 57)(8 58)(9 59)(10 60)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)
(6 56)(7 57)(8 58)(9 59)(10 60)(16 21)(17 22)(18 23)(19 24)(20 25)(26 31)(27 32)(28 33)(29 34)(30 35)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 51 41)(7 52 42)(8 53 43)(9 54 44)(10 55 45)(11 31 21)(12 32 22)(13 33 23)(14 34 24)(15 35 25)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(2 5)(3 4)(6 55)(7 54)(8 53)(9 52)(10 51)(11 15)(12 14)(16 30)(17 29)(18 28)(19 27)(20 26)(21 35)(22 34)(23 33)(24 32)(25 31)(36 40)(37 39)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)

G:=sub<Sym(60)| (1,43,13,38)(2,44,14,39)(3,45,15,40)(4,41,11,36)(5,42,12,37)(6,31,56,26)(7,32,57,27)(8,33,58,28)(9,34,59,29)(10,35,60,30)(16,51,21,46)(17,52,22,47)(18,53,23,48)(19,54,24,49)(20,55,25,50), (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), (1,13)(2,14)(3,15)(4,11)(5,12)(6,56)(7,57)(8,58)(9,59)(10,60)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45), (6,56)(7,57)(8,58)(9,59)(10,60)(16,21)(17,22)(18,23)(19,24)(20,25)(26,31)(27,32)(28,33)(29,34)(30,35)(46,51)(47,52)(48,53)(49,54)(50,55), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,51,41)(7,52,42)(8,53,43)(9,54,44)(10,55,45)(11,31,21)(12,32,22)(13,33,23)(14,34,24)(15,35,25)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (2,5)(3,4)(6,55)(7,54)(8,53)(9,52)(10,51)(11,15)(12,14)(16,30)(17,29)(18,28)(19,27)(20,26)(21,35)(22,34)(23,33)(24,32)(25,31)(36,40)(37,39)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)>;

G:=Group( (1,43,13,38)(2,44,14,39)(3,45,15,40)(4,41,11,36)(5,42,12,37)(6,31,56,26)(7,32,57,27)(8,33,58,28)(9,34,59,29)(10,35,60,30)(16,51,21,46)(17,52,22,47)(18,53,23,48)(19,54,24,49)(20,55,25,50), (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), (1,13)(2,14)(3,15)(4,11)(5,12)(6,56)(7,57)(8,58)(9,59)(10,60)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45), (6,56)(7,57)(8,58)(9,59)(10,60)(16,21)(17,22)(18,23)(19,24)(20,25)(26,31)(27,32)(28,33)(29,34)(30,35)(46,51)(47,52)(48,53)(49,54)(50,55), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,51,41)(7,52,42)(8,53,43)(9,54,44)(10,55,45)(11,31,21)(12,32,22)(13,33,23)(14,34,24)(15,35,25)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (2,5)(3,4)(6,55)(7,54)(8,53)(9,52)(10,51)(11,15)(12,14)(16,30)(17,29)(18,28)(19,27)(20,26)(21,35)(22,34)(23,33)(24,32)(25,31)(36,40)(37,39)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56) );

G=PermutationGroup([[(1,43,13,38),(2,44,14,39),(3,45,15,40),(4,41,11,36),(5,42,12,37),(6,31,56,26),(7,32,57,27),(8,33,58,28),(9,34,59,29),(10,35,60,30),(16,51,21,46),(17,52,22,47),(18,53,23,48),(19,54,24,49),(20,55,25,50)], [(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)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,56),(7,57),(8,58),(9,59),(10,60),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45)], [(6,56),(7,57),(8,58),(9,59),(10,60),(16,21),(17,22),(18,23),(19,24),(20,25),(26,31),(27,32),(28,33),(29,34),(30,35),(46,51),(47,52),(48,53),(49,54),(50,55)], [(1,28,18),(2,29,19),(3,30,20),(4,26,16),(5,27,17),(6,51,41),(7,52,42),(8,53,43),(9,54,44),(10,55,45),(11,31,21),(12,32,22),(13,33,23),(14,34,24),(15,35,25),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(2,5),(3,4),(6,55),(7,54),(8,53),(9,52),(10,51),(11,15),(12,14),(16,30),(17,29),(18,28),(19,27),(20,26),(21,35),(22,34),(23,33),(24,32),(25,31),(36,40),(37,39),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56)]])

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E ··· 4J 5A 5B 6 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 ··· 4 5 5 6 10 10 10 10 10 10 12 12 15 15 15 15 20 20 20 20 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 3 3 30 30 8 1 1 3 3 30 ··· 30 2 2 8 2 2 6 6 6 6 8 8 8 8 8 8 2 2 2 2 6 6 6 6 8 8 8 8 8 ··· 8

52 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 6 6 6 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D5 D6 D10 C4×S3 D15 C4×D5 D30 C4×D15 S4 C2×S4 C4×S4 C5⋊S4 C2×C5⋊S4 C4×C5⋊S4 kernel C4×C5⋊S4 A4⋊Dic5 A4×C20 C2×C5⋊S4 C5⋊S4 C22×C20 C4×A4 C22×C10 C2×A4 C2×C10 C22×C4 A4 C23 C22 C20 C10 C5 C4 C2 C1 # reps 1 1 1 1 4 1 2 1 2 2 4 4 4 8 2 2 4 2 2 4

Matrix representation of C4×C5⋊S4 in GL5(𝔽61)

 50 0 0 0 0 0 50 0 0 0 0 0 50 0 0 0 0 0 50 0 0 0 0 0 50
,
 17 60 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 60
,
 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
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 17 60 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,50],[17,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,60],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,17,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C4×C5⋊S4 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes S_4
% in TeX

G:=Group("C4xC5:S4");
// GroupNames label

G:=SmallGroup(480,1025);
// by ID

G=gap.SmallGroup(480,1025);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,36,451,3364,10085,1286,5886,2232]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^5=c^2=d^2=e^3=f^2=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,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

׿
×
𝔽