C₂×C₈²×C₃₇ #17990265

order	4736
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 884736
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₈

Aliases

abelian

C₂×C₈²×C₃₇

Subgroups

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

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 55 points:

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

Conjugacy classes & character table

4736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7136, C₃₇×#7138, C₃₇×#7151, …
maximal normal subgroup of	C₃₇×#7136, C₃₇×#7138, C₃₇×#7151, …
maximal quotient of	C₂×C₃₇×#1273, C₃₇×#8131, C₈×C₂²⋊C₈×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 179);

C2×C82×C37 #17990265