C₃₇×#532 #14557197

order	2368
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₇×#532

Subgroups

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

310 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 1184

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)(42,43)(44,47)(45,46)(48,49);(43,46)(47,49);(38,39,40,41);(42,44,45,48)(43,47,46,49)

Conjugacy classes & character table

1036 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1480, C₃₇×#1484, C₃₇×#1492, …
maximal normal subgroup of	C₃₇×#1480, C₃₇×#1484, C₃₇×#1492, …
maximal quotient of	C₃₇×#2285, C₃₇×#2293, C₃₇×#2297, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 213);