2₋¹⁺⁴⋊C₅ #4124

order	160
automorphism	Aut(G) = #66297 (order 320) · Inn(G) = C₂⁴⋊C₅ · Out(G) = C₄
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

2₋¹⁺⁴⋊C₅

Splittings

Fitting, derived, Sylow

2₋¹⁺⁴ ⋊ C₅

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	C₂⁴⋊C₅
commutator, Fitting, cosocle, hypocenter	2₋¹⁺⁴	C₅
LCS	G ▷ 2₋¹⁺⁴
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(2,5,20,8,11)(3,9,24,13,17)(4,15,29,25,31)(7,10,22,32,16)(12,14,21,23,28)(18,19,26,27,30);(1,21,6,9)(2,22,10,5)(3,15,14,26)(4,25,19,30)(7,13,11,28)(8,18,16,31)(12,32,17,20)(23,29,24,27)

Polycyclic presentation:

5,2^5|c^2=f,d^2=f,b^a=e,c^a=be,d^a=bce,e^a=bcde,c^b=cf,d^b=df,e^b=ef,d^c=df

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#66243, #66244, #10617016, …
maximal subgroup of	#66243, #66244, C₂×2₋¹⁺⁴⋊C₅, …
maximal normal subgroup of	#66243, #66244, C₂×2₋¹⁺⁴⋊C₅, …
maximal quotient of	#564⋊C₅, C₂×2₋¹⁺⁴⋊C₅, C₃×2₋¹⁺⁴⋊C₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 199);

2−1+4⋊C5 #4124