C₂²×C₂₃×H(5,𝔽₂) #15671210

order	2944
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₂₃ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×C₂₃×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₂₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

6560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 6 generators on 35 points:

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

Conjugacy classes & character table

1564 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₃×#20585, C₂₃×#20586, C₂₃×#20592, …
maximal normal subgroup of	C₂₃×#20585, C₂₃×#20586, C₂₃×#20592, …
maximal quotient of	C₂²×C₂₃×#518, C₂²×C₂₃×#520, C₂×C₄×C₂₃×H(5,𝔽₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 2323);

C22×C23×H(5,𝔽2) #15671210