C₄×#126×C₃₇ #17991122

order	4736
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×#126×C₃₇

Subgroups

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

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 57
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 57 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(6,10)(7,14)(12,19)(13,17)(15,18)(16,20);(5,6,11,17)(7,12,18,19)(8,10,9,13)(14,20,15,16);(5,7,8,15)(6,12,10,16)(9,14,11,18)(13,20,17,19)

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₈×#126×C₃₇, C₃₇×#21665, C₃₇×#21666, …
maximal normal subgroup of	C₈×#126×C₃₇, C₃₇×#21665, C₃₇×#21666, …
maximal quotient of	C₄×C₃₇×#388, C₄×C₃₇×#387, C₄×C₃₇×#383, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1036);