C₃₇×#535 #14557200

order	2368
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃₇×#535

Subgroups

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

334 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 592

A minimal faithful permutation representation — 4 generators on 49 points:

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

Conjugacy classes & character table

814 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#1910, C₃₇×#1911, C₃₇×#1914, …
maximal normal subgroup of	C₃₇×#1910, C₃₇×#1911, C₃₇×#1914, …
maximal quotient of	C₃₇×#2309, C₃₇×#2311, C₃₇×#2317, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 216);