C2^3xF5 - GroupNames

G = C₂³×F₅ order 160 = 2⁵·5

Direct product of C₂³ and F₅

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂³×F₅, D₅.C₂⁴, D₁₀.₁₅C₂³, C₅⋊(C₂³×C₄), C₁₀⋊(C₂²×C₄), D₅⋊(C₂²×C₄), D₁₀⋊₉(C₂×C₄), (C₂²×C₁₀)⋊₅C₄, (C₂²×D₅)⋊₆C₄, (C₂³×D₅).₄C₂, (C₂²×D₅).₄₀C₂², (C₂×C₁₀)⋊₃(C₂×C₄), SmallGroup(160,236)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂³×F₅

Chief series

C₁ — C₅ — D₅ — F₅ — C₂×F₅ — C₂²×F₅ — C₂³×F₅

Lower central

C₅ — C₂³×F₅

Upper central

C₁ — C₂³

Generators and relations for C₂³×F₅
G = < a,b,c,d,e | a²=b²=c²=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 644 in 236 conjugacy classes, 134 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂³, C₂³, D₅, D₅, C₁₀, C₂²×C₄, C₂⁴, F₅, D₁₀, C₂×C₁₀, C₂³×C₄, C₂×F₅, C₂²×D₅, C₂²×C₁₀, C₂²×F₅, C₂³×D₅, C₂³×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, F₅, C₂³×C₄, C₂×F₅, C₂²×F₅, C₂³×F₅

Smallest permutation representation of C₂³×F₅
►On 40 points

Generators in S₄₀

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

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

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

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

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

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

C₂³×F₅ is a maximal subgroup of D₁₀⋊(C₄⋊C₄) (C₂×F₅)⋊D₄
C₂³×F₅ is a maximal quotient of Dic₅.C₂⁴ D₁₀.C₂⁴ Dic₅.₂₀C₂⁴ D₅.2_-¹⁺⁴ Dic₅.₂₁C₂⁴ Dic₅.₂₂C₂⁴ D₅.2₊¹⁺⁴

40 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4P	5	10A	···	10G
order	1	2	···	2	2	···	2	4	···	4	5	10	···	10
size	1	1	···	1	5	···	5	5	···	5	4	4	···	4

40 irreducible representations

dim	1	1	1	1	1	4	4
type	+	+	+			+	+
image	C₁	C₂	C₂	C₄	C₄	F₅	C₂×F₅
kernel	C₂³×F₅	C₂²×F₅	C₂³×D₅	C₂²×D₅	C₂²×C₁₀	C₂³	C₂²
# reps	1	14	1	14	2	1	7

Matrix representation of C₂³×F₅ ►in GL₇(𝔽₄₁)

1	0	0	0	0	0	0
0	40	0	0	0	0	0
0	0	40	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

40	0	0	0	0	0	0
0	40	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

40	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	0	0	40
0	0	0	1	0	0	40
0	0	0	0	1	0	40
0	0	0	0	0	1	40

32	0	0	0	0	0	0
0	32	0	0	0	0	0
0	0	9	0	0	0	0
0	0	0	0	0	1	0
0	0	0	1	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0

G:=sub<GL(7,GF(41))| [1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,40,40,40,40],[32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

C₂³×F₅ in GAP, Magma, Sage, TeX

C_2^3\times F_5

% in TeX

G:=Group("C2^3xF5");

// GroupNames label

G:=SmallGroup(160,236);

// by ID

G=gap.SmallGroup(160,236);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,2309,317]);

// Polycyclic

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

// generators/relations

G = C23×F5 order 160 = 25·5

Direct product of C23 and F5

G = C₂³×F₅ order 160 = 2⁵·5

Direct product of C₂³ and F₅