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G = C244F5order 320 = 26·5

1st semidirect product of C24 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C244F5, C5⋊(C243C4), (C23×C10)⋊7C4, (C23×D5)⋊11C4, D5.3C22≀C2, (D5×C24).4C2, C23.52(C2×F5), D105(C22⋊C4), D10.102(C2×D4), C222(C22⋊F5), (C22×F5)⋊1C22, (C22×D5).149D4, C22.101(C22×F5), (C22×D5).281C23, (C23×D5).136C22, (C2×C22⋊F5)⋊6C2, (C2×C10)⋊2(C22⋊C4), C2.41(C2×C22⋊F5), C10.41(C2×C22⋊C4), (C22×C10).76(C2×C4), (C2×C10).94(C22×C4), (C22×D5).130(C2×C4), SmallGroup(320,1138)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C244F5
C1C5D5D10C22×D5C22×F5C2×C22⋊F5 — C244F5
C5C2×C10 — C244F5
C1C22C24

Generators and relations for C244F5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f4=1, ab=ba, ac=ca, faf-1=ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 2426 in 506 conjugacy classes, 80 normal (9 characteristic)
C1, C2 [×3], C2 [×16], C4 [×4], C22, C22 [×6], C22 [×80], C5, C2×C4 [×12], C23 [×3], C23 [×84], D5 [×4], D5 [×6], C10 [×3], C10 [×6], C22⋊C4 [×12], C22×C4 [×4], C24, C24 [×18], F5 [×4], D10 [×12], D10 [×54], C2×C10, C2×C10 [×6], C2×C10 [×14], C2×C22⋊C4 [×6], C25, C2×F5 [×12], C22×D5, C22×D5 [×15], C22×D5 [×62], C22×C10 [×3], C22×C10 [×6], C243C4, C22⋊F5 [×12], C22×F5 [×4], C23×D5 [×6], C23×D5 [×12], C23×C10, C2×C22⋊F5 [×6], D5×C24, C244F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], F5, C2×C22⋊C4 [×3], C22≀C2 [×4], C2×F5 [×3], C243C4, C22⋊F5 [×6], C22×F5, C2×C22⋊F5 [×3], C244F5

Smallest permutation representation of C244F5
On 40 points
Generators in S40
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(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)
(1 23)(2 25 5 21)(3 22 4 24)(6 28)(7 30 10 26)(8 27 9 29)(11 33)(12 35 15 31)(13 32 14 34)(16 38)(17 40 20 36)(18 37 19 39)

G:=sub<Sym(40)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,23)(2,25,5,21)(3,22,4,24)(6,28)(7,30,10,26)(8,27,9,29)(11,33)(12,35,15,31)(13,32,14,34)(16,38)(17,40,20,36)(18,37,19,39)>;

G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,23)(2,25,5,21)(3,22,4,24)(6,28)(7,30,10,26)(8,27,9,29)(11,33)(12,35,15,31)(13,32,14,34)(16,38)(17,40,20,36)(18,37,19,39) );

G=PermutationGroup([(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)], [(1,23),(2,25,5,21),(3,22,4,24),(6,28),(7,30,10,26),(8,27,9,29),(11,33),(12,35,15,31),(13,32,14,34),(16,38),(17,40,20,36),(18,37,19,39)])

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M2N···2S4A···4H 5 10A···10O
order12222···222222···24···4510···10
size11112···2555510···1020···2044···4

44 irreducible representations

dim111112444
type+++++++
imageC1C2C2C4C4D4F5C2×F5C22⋊F5
kernelC244F5C2×C22⋊F5D5×C24C23×D5C23×C10C22×D5C24C23C22
# reps16162121312

Matrix representation of C244F5 in GL6(𝔽41)

100000
0400000
001000
000100
000010
000001
,
100000
0400000
001000
000100
0000400
0000040
,
4000000
0400000
0040000
0004000
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
0004000
001600
00004035
0000635
,
090000
900000
000010
000001
0063500
00403500

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,1,0,0,0,0,0,0,1,0,0] >;

C244F5 in GAP, Magma, Sage, TeX

C_2^4\rtimes_4F_5
% in TeX

G:=Group("C2^4:4F5");
// GroupNames label

G:=SmallGroup(320,1138);
// by ID

G=gap.SmallGroup(320,1138);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,422,6278,1595]);
// Polycyclic

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

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