Copied to
clipboard

## G = C5×C22⋊A4order 240 = 24·3·5

### Direct product of C5 and C22⋊A4

Aliases: C5×C22⋊A4, C244C15, (C2×C10)⋊A4, C22⋊(C5×A4), (C23×C10)⋊2C3, SmallGroup(240,204)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C5×C22⋊A4
 Chief series C1 — C22 — C24 — C23×C10 — C5×C22⋊A4
 Lower central C24 — C5×C22⋊A4
 Upper central C1 — C5

Generators and relations for C5×C22⋊A4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 208 in 68 conjugacy classes, 16 normal (6 characteristic)
C1, C2, C3, C22, C22, C5, C23, C10, A4, C15, C24, C2×C10, C2×C10, C22×C10, C22⋊A4, C5×A4, C23×C10, C5×C22⋊A4
Quotients: C1, C3, C5, A4, C15, C22⋊A4, C5×A4, C5×C22⋊A4

Smallest permutation representation of C5×C22⋊A4
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)(6 20)(7 16)(8 17)(9 18)(10 19)(11 57)(12 58)(13 59)(14 60)(15 56)(26 34)(27 35)(28 31)(29 32)(30 33)
(6 20)(7 16)(8 17)(9 18)(10 19)(11 57)(12 58)(13 59)(14 60)(15 56)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 14)(7 15)(8 11)(9 12)(10 13)(16 56)(17 57)(18 58)(19 59)(20 60)(26 34)(27 35)(28 31)(29 32)(30 33)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 20)(7 16)(8 17)(9 18)(10 19)(11 57)(12 58)(13 59)(14 60)(15 56)(21 27)(22 28)(23 29)(24 30)(25 26)(36 47)(37 48)(38 49)(39 50)(40 46)(41 55)(42 51)(43 52)(44 53)(45 54)
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 54 34)(7 55 35)(8 51 31)(9 52 32)(10 53 33)(11 42 22)(12 43 23)(13 44 24)(14 45 25)(15 41 21)(16 47 27)(17 48 28)(18 49 29)(19 50 30)(20 46 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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(26,34)(27,35)(28,31)(29,32)(30,33), (6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53), (1,21)(2,22)(3,23)(4,24)(5,25)(6,14)(7,15)(8,11)(9,12)(10,13)(16,56)(17,57)(18,58)(19,59)(20,60)(26,34)(27,35)(28,31)(29,32)(30,33)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46), (1,35)(2,31)(3,32)(4,33)(5,34)(6,20)(7,16)(8,17)(9,18)(10,19)(11,57)(12,58)(13,59)(14,60)(15,56)(21,27)(22,28)(23,29)(24,30)(25,26)(36,47)(37,48)(38,49)(39,50)(40,46)(41,55)(42,51)(43,52)(44,53)(45,54), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,54,34)(7,55,35)(8,51,31)(9,52,32)(10,53,33)(11,42,22)(12,43,23)(13,44,24)(14,45,25)(15,41,21)(16,47,27)(17,48,28)(18,49,29)(19,50,30)(20,46,26)>;

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

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

C5×C22⋊A4 is a maximal subgroup of   (C22×D5)⋊A4  C244D15

40 conjugacy classes

 class 1 2A ··· 2E 3A 3B 5A 5B 5C 5D 10A ··· 10T 15A ··· 15H order 1 2 ··· 2 3 3 5 5 5 5 10 ··· 10 15 ··· 15 size 1 3 ··· 3 16 16 1 1 1 1 3 ··· 3 16 ··· 16

40 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C3 C5 C15 A4 C5×A4 kernel C5×C22⋊A4 C23×C10 C22⋊A4 C24 C2×C10 C22 # reps 1 2 4 8 5 20

Matrix representation of C5×C22⋊A4 in GL6(𝔽31)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 30
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30
,
 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 30
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(31))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,30],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C5×C22⋊A4 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes A_4
% in TeX

G:=Group("C5xC2^2:A4");
// GroupNames label

G:=SmallGroup(240,204);
// by ID

G=gap.SmallGroup(240,204);
# by ID

G:=PCGroup([6,-3,-5,-2,2,-2,2,542,1083,3604,6485]);
// Polycyclic

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

׿
×
𝔽