#65334

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #485

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
hypercenter	C₂²⋊Q₈	D₅
hypocenter	C₅	#485
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 29 points:

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

Polycyclic presentation:

2^6,5|a^2=e,b^2=f,c^2=f,d^2=f,c^a=cd,d^a=df,g^a=g^4,c^b=ce,d^c=df

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10611867, #10611868, #10611885, …
maximal normal subgroup of	#10611867, #10611868, #10611885, …
maximal quotient of	#10600296, #10600299, #10600311, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 672);