C₃₇×#575 #14557240

order	2368
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₇×#575

Subgroups

	subgroup	G/H
center	C₄×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃₇
socle	C₂×C₃₇	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 592

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)(40,47)(41,46)(42,50)(43,45)(44,48)(49,53)(51,52);(38,40)(39,44)(41,48)(42,51)(43,49)(45,52)(46,53)(47,50);(39,45)(44,52)(46,50)(47,53);(38,41,43,42)(39,46,45,50)(40,48,49,51)(44,53,52,47)

Conjugacy classes & character table

814 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#1881, C₃₇×#1906, C₃₇×#2008, …
maximal normal subgroup of	C₃₇×#1881, C₃₇×#1906, C₃₇×#2008, …
maximal quotient of	C₃₇×#2896, C₃₇×#2911, C₃₇×#2943, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 256);