C₃₇×#554 #14557219

order	2368
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#554

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃₇
prime radical	C₂×D₄×C₃₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ G

228 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 53
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 53 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)(38,39,42,41)(40,44,45,43)(46,47,50,49)(48,52,53,51);(39,41)(43,44)(46,48,50,53)(47,51,49,52);(38,40)(39,43)(41,44)(42,45);(38,41,42,39)(40,44,45,43)(46,49,50,47)(48,51,53,52)

Conjugacy classes & character table

925 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#1622, C₃₇×#1632, C₃₇×#1643, …
maximal normal subgroup of	C₃₇×#1622, C₃₇×#1632, C₃₇×#1643, …
maximal quotient of	C₃₇×#2345, D₄×C₄⋊C₄×C₃₇, C₃₇×#2349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 235);