C₄×C₂₃×H(5,𝔽₂) #15671048

order	2944
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₂₃ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₄×C₂₃×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂⁴
commutator	C₂	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂²	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁵
prime radical	C₂×C₂₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

1660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 35 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35);(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×#8316, C₂₃×#8318, C₂₃×#17438, …
maximal normal subgroup of	C₂₃×#8316, C₂₃×#8318, C₂₃×#17438, …
maximal quotient of	C₄×C₂₃×#518, C₄×C₂₃×#520, C₄×C₂₃×#534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 2161);

C4×C23×H(5,𝔽2) #15671048