C2^4:4F5 - GroupNames

G = C₂⁴⋊₄F₅ order 320 = 2⁶·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴⋊₄F₅, C₅⋊(C₂⁴⋊₃C₄), (C₂³×C₁₀)⋊₇C₄, (C₂³×D₅)⋊₁₁C₄, D₅.₃C₂²≀C₂, (D₅×C₂⁴).₄C₂, C₂³.₅₂(C₂×F₅), D₁₀⋊₅(C₂²⋊C₄), D₁₀.₁₀₂(C₂×D₄), C₂²⋊₂(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

Derived series

C₁ — C₂×C₁₀ — C₂⁴⋊₄F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²×D₅ — C₂²×F₅ — C₂×C₂²⋊F₅ — C₂⁴⋊₄F₅

Lower central

C₅ — C₂×C₁₀ — C₂⁴⋊₄F₅

Upper central

C₁ — C₂² — C₂⁴

Generators and relations for C₂⁴⋊₄F₅
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, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂³, C₂³, D₅, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂⁴, C₂⁴, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₂⁵, C₂×F₅, C₂²×D₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂⁴⋊₃C₄, C₂²⋊F₅, C₂²×F₅, C₂³×D₅, C₂³×D₅, C₂³×C₁₀, C₂×C₂²⋊F₅, D₅×C₂⁴, C₂⁴⋊₄F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, F₅, C₂×C₂²⋊C₄, C₂²≀C₂, C₂×F₅, C₂⁴⋊₃C₄, C₂²⋊F₅, C₂²×F₅, C₂×C₂²⋊F₅, C₂⁴⋊₄F₅

Smallest permutation representation of C₂⁴⋊₄F₅
►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	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	2N	···	2S	4A	···	4H	5	10A	···	10O
order	1	2	2	2	2	···	2	2	2	2	2	2	···	2	4	···	4	5	10	···	10
size	1	1	1	1	2	···	2	5	5	5	5	10	···	10	20	···	20	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	2	4	4	4
type	+	+	+			+	+	+	+
image	C₁	C₂	C₂	C₄	C₄	D₄	F₅	C₂×F₅	C₂²⋊F₅
kernel	C₂⁴⋊₄F₅	C₂×C₂²⋊F₅	D₅×C₂⁴	C₂³×D₅	C₂³×C₁₀	C₂²×D₅	C₂⁴	C₂³	C₂²
# reps	1	6	1	6	2	12	1	3	12

Matrix representation of C₂⁴⋊₄F₅ ►in GL₆(𝔽₄₁)

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	40	0	0
0	0	1	6	0	0
0	0	0	0	40	35
0	0	0	0	6	35

0	9	0	0	0	0
9	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	6	35	0	0
0	0	40	35	0	0

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] >;

C₂⁴⋊₄F₅ 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

G = C24⋊4F5 order 320 = 26·5

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

G = C₂⁴⋊₄F₅ order 320 = 2⁶·5

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