C₂²×M₃₂×C₃₇ #17992223

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×M₃₂×C₃₇

Subgroups

	subgroup	G/H
center	C₂²×C₈×C₃₇	C₂²
commutator	C₂	C₂³×C₈×C₃₇
Frattini, cosocle	C₈	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂×C₈
prime radical	C₂⁴×C₃₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₇ ◁ G

400 conjugacy classes of subgroups.

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)(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);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,14)(8,16)(10,18)(12,20)

Conjugacy classes & character table

2960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12544, C₃₇×#12545, C₂×C₃₇×#1375, …
maximal normal subgroup of	C₃₇×#12544, C₃₇×#12545, C₂×C₃₇×#1375, …
maximal quotient of	C₂²×C₃₇×#346, C₂×C₄×M₃₂×C₃₇, C₂²×C₃₇×C₂²⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2137);

C22×M32×C37 #17992223