C₂×C₄×#41×C₃₇ #17992242

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×C₄×#41×C₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₃₇	C₂²
commutator	C₂	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁵×C₃₇
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂²×#41×C₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₇ ◁ G

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 5 generators on 51 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8312, C₃₇×#10758, C₃₇×#10772, …
maximal normal subgroup of	C₃₇×#8312, C₃₇×#10758, C₃₇×#10772, …
maximal quotient of	C₂×C₄×#117×C₃₇, C₂×C₄²×D₄×C₃₇, C₂×C₄²×Q₈×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2156);

C2×C4×#41×C37 #17992242