C₂×H(5,𝔽₂)×C₃₇ #14557248

order	2368
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×H(5,𝔽₂)×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₃₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ G

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 592

A minimal faithful permutation representation — 5 generators on 47 points:

(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(3,4)(5,9)(6,7)(8,10);(4,7)(8,9);(3,4)(5,8)(6,7)(9,10);(3,5)(4,8)(6,10)(7,9)

Conjugacy classes & character table

1258 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1790, C₃₇×#1792, C₃₇×#1795, …
maximal normal subgroup of	C₃₇×#1790, C₃₇×#1792, C₃₇×#1795, …
maximal quotient of	C₂×C₃₇×#518, C₂×C₃₇×#520, C₄×H(5,𝔽₂)×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 264);

C2×H(5,𝔽2)×C37 #14557248