C₃₇×#1790 #17990608

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₇×#1790

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator	C₂³	C₂²×C₄×C₃₇
Frattini, cosocle	C₂⁴	C₂³×C₃₇
prime radical	C₂×H(5,𝔽₂)×C₃₇	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂⁴×C₃₇ ◁ G

670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 3 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)(38,39)(40,41)(42,43)(44,48,49,46)(45,47);(42,44)(43,46)(45,49)(47,48);(39,41)(42,43)(44,46)(45,47)(48,49)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#13041, C₃₇×#13042, C₃₇×#13043, …
maximal normal subgroup of	C₃₇×#13041, C₃₇×#13042, C₃₇×#13043, …
maximal quotient of	C₃₇×#8316, C₃₇×#8317, C₃₇×#8346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 522);