C₂×C₄×C₄⋊C₄×C₃₇ #17991087

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₃₇ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×C₄×C₄⋊C₄×C₃₇

Subgroups

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

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 4 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);(8,10)(11,12,13,14);(7,8,9,10)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₇×#1279, C₃₇×#7892, C₃₇×#7901, …
maximal normal subgroup of	C₂×C₃₇×#1279, C₃₇×#7892, C₃₇×#7901, …
maximal quotient of	C₂×C₄×#95×C₃₇, C₄²×C₄⋊C₄×C₃₇, C₂×C₃₇×#1434, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1001);

C2×C4×C4⋊C4×C37 #17991087