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G = C5×F8  order 280 = 23·5·7

Direct product of C5 and F8

direct product, metabelian, soluble, monomial, A-group

Aliases: C5×F8, C23⋊C35, (C22×C10)⋊C7, SmallGroup(280,33)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C5×F8
Chief series
C1 — C23 — F8 — C5×F8
Lower central
C23 — C5×F8
Upper central
C1 — C5

Generators and relations for C5×F8
 G = < a,b,c,d,e | a5=b2=c2=d2=e7=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=dc=cd, ece-1=b, ede-1=c >

C1
7C2
C5
8C7
7C22
7C10
8C35
C23
7C2×C10
C22×C10
F8
C5×F8

Smallest permutation representation of C5×F8
►On 40 points
Generators in S40
(1 3 2 4 5)(6 35 29 25 14)(7 36 30 26 15)(8 37 31 20 16)(9 38 32 21 17)(10 39 33 22 18)(11 40 27 23 19)(12 34 28 24 13)

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

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

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

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

40 conjugacy classes

class 1  2 5A5B5C5D7A···7F10A10B10C10D35A···35X
order1255557···71010101035···35
size1711118···877778···8

40 irreducible representations

dim111177
type++
imageC1C5C7C35F8C5×F8
kernelC5×F8F8C22×C10C23C5C1
# reps1462414

Matrix representation of C5×F8 ►in GL8(𝔽71)

540000000
01000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
00000001
070707070707070
00000100
00000010
00010000
00001000
01000000
,
10000000
00001000
00010000
00100000
01000000
070707070707070
00000001
00000010
,
10000000
00000100
00000010
00000001
070707070707070
01000000
00100000
00010000
,
370000000
01000000
00010000
00000100
00000001
00001000
00100000
070707070707070

G:=sub<GL(8,GF(71))| [54,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,70,0,0,0,0,1,0,0,70,0,0,0,0,0,0,0,70,0,0,1,0,0,0,0,70,0,0,0,1,0,0,0,70,1,0,0,0,0,0,0,70,0,1,0,0,0,0,1,70,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,1,70,0,0,0,0,0,1,0,70,0,0,0,0,1,0,0,70,0,0,0,1,0,0,0,70,0,0,0,0,0,0,0,70,0,0,0,0,0,0,0,70,0,1,0,0,0,0,0,70,1,0],[1,0,0,0,0,0,0,0,0,0,0,0,70,1,0,0,0,0,0,0,70,0,1,0,0,0,0,0,70,0,0,1,0,0,0,0,70,0,0,0,0,1,0,0,70,0,0,0,0,0,1,0,70,0,0,0,0,0,0,1,70,0,0,0],[37,0,0,0,0,0,0,0,0,1,0,0,0,0,0,70,0,0,0,0,0,0,1,70,0,0,1,0,0,0,0,70,0,0,0,0,0,1,0,70,0,0,0,1,0,0,0,70,0,0,0,0,0,0,0,70,0,0,0,0,1,0,0,70] >;
 

C5×F8 in GAP, Magma, Sage, TeX 

C_5\times F_8
 
% in TeX
 
G:=Group("C5xF8");
 
// GroupNames label
 
G:=SmallGroup(280,33);
 
// by ID
 
G=gap.SmallGroup(280,33);
 
# by ID
 
G:=PCGroup([5,-5,-7,-2,2,2,217,568,884]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^7=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=d*c=c*d,e*c*e^-1=b,e*d*e^-1=c>;
 
// generators/relations
 

Export

Subgroup lattice of C5×F8 in TeX

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