C₃₇×#3395 #17992213

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3395

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄×Q₈×C₃₇ ◁ G

348 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 61
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 4 generators on 61 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)(39,43)(44,45)(46,47,50,53)(48,54,56,61)(49,57,59,52)(51,58,60,55);(38,39,42,43)(40,44,41,45);(38,40,42,41)(39,44,43,45)(46,48,50,56)(47,51,53,60)(49,58,59,55)(52,61,57,54);(38,41,42,40)(39,44,43,45)

Conjugacy classes & character table

1295 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12330, C₃₇×#12334, C₃₇×#12382, …
maximal normal subgroup of	C₃₇×#12330, C₃₇×#12334, C₃₇×#12382, …
maximal quotient of	C₃₇×#22542, Q₈×#107×C₃₇, C₃₇×#22808, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2127);