C₂²×C₄×D₄×C₃₇ #17992240

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₃₇ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×C₄×D₄×C₃₇

Subgroups

	subgroup	G/H
center	C₂³×C₄×C₃₇	C₂²
commutator	C₂	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁵×C₃₇
socle	C₂⁴×C₃₇	C₂³
prime radical	C₂³×D₄×C₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₇ ◁ G

2528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 5 generators on 49 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8301, C₃₇×#8302, C₃₇×#8310, …
maximal normal subgroup of	C₃₇×#8301, C₃₇×#8302, C₃₇×#8310, …
maximal quotient of	C₂²×C₄×C₂²⋊C₄×C₃₇, C₂²×C₄×C₄⋊C₄×C₃₇, C₂×C₄²×D₄×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2154);

C22×C4×D4×C37 #17992240