C₄×#103×C₃₇ #17990579

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×#103×C₃₇

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₇	D₄
commutator	C₄	C₂×C₄²×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle, prime radical	C₂³×C₃₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

332 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 53 points:

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

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₈×#103×C₃₇, C₃₇×#8201, C₃₇×#8203, …
maximal normal subgroup of	C₈×#103×C₃₇, C₃₇×#8201, C₃₇×#8203, …
maximal quotient of	C₄×C₃₇×Q₈⋊C₈, C₃₇×#8175, C₃₇×#8176, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 493);