C₄×H(5,𝔽₂)×C₃₇ #17992247

order	4736
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₄×H(5,𝔽₂)×C₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂⁴
commutator	C₂	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁵×C₃₇
socle	C₂²×C₃₇	C₂⁵
prime radical	C₂×H(5,𝔽₂)×C₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ G

1660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 5 generators on 49 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(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

2516 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8316, C₃₇×#8318, C₃₇×#17438, …
maximal normal subgroup of	C₃₇×#8316, C₃₇×#8318, C₃₇×#17438, …
maximal quotient of	C₄×C₃₇×#518, C₄×C₃₇×#520, C₄×C₃₇×#534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2161);

C4×H(5,𝔽2)×C37 #17992247