#65706

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #433

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₅×C₄⋊C₄	C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#433
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#65706, #10617078, #10617079, …
maximal normal subgroup of	C₂×#65706, #10617078, #10617079, …
maximal quotient of	#10603774, #10603778, #10603782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1044);