C₄²×D₄×C₃₇ #17991089

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

C₄²×D₄×C₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₃₇	C₂²
commutator	C₂	C₂²×C₄²×C₃₇
Frattini, cosocle	C₂³	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂²×D₄×C₃₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₇ ◁ G

924 conjugacy classes of subgroups; only the 576 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 576 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) = 49
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 49 points:

(1,2,3,4)(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);(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	C₄×C₃₇×D₄⋊C₈, C₃₇×#8170, C₃₇×#8172, …
maximal normal subgroup of	C₄×C₃₇×D₄⋊C₈, C₃₇×#8170, C₃₇×#8172, …
maximal quotient of	C₄²×C₂²⋊C₄×C₃₇, C₄²×C₄⋊C₄×C₃₇, C₄×C₃₇×#386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1003);