C₂²×C₂₉×H(5,𝔽₂) #16817305

order	3712
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₂₉ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×C₂₉×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₂₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

6560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 6 generators on 41 points:

(1,2)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(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

1972 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#20585, C₂₉×#20586, C₂₉×#20592, …
maximal normal subgroup of	C₂₉×#20585, C₂₉×#20586, C₂₉×#20592, …
maximal quotient of	C₂²×C₂₉×#518, C₂²×C₂₉×#520, C₂×C₄×C₂₉×H(5,𝔽₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2323);

C22×C29×H(5,𝔽2) #16817305