C₂×C₃₇×#520 #17992245

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

C₂×C₃₇×#520

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator	C₂	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁵×C₃₇
prime radical	C₂²×#41×C₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

1640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 5 generators on 51 points:

(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(3,4,5,6)(7,8)(9,13)(10,11)(12,14);(8,11)(12,13);(7,8)(9,12)(10,11)(13,14);(7,9,10,14)(8,12,11,13)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8324, C₃₇×#10765, C₃₇×#10775, …
maximal normal subgroup of	C₃₇×#8324, C₃₇×#10765, C₃₇×#10775, …
maximal quotient of	C₂×C₃₇×#2274, C₂×C₃₇×#2275, C₂×C₃₇×#2276, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2159);

C2×C37×#520 #17992245