C₃₇×SD₁₆⋊C₈ #17990396

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃₇×SD₁₆⋊C₈
unambiguous semidirect product	SD₁₆⋊(C₈×C₃₇)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₄
commutator	C₄	C₂²×C₈×C₃₇
Frattini, cosocle	C₄²	C₂³×C₃₇
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂×D₄×C₃₇	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₄×C₈×C₃₇ ◁ G

176 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 2368

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

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

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×SD₁₆⋊C₈, C₃₇×#17849, C₈×AGL(1,ℤ/8)×C₃₇, …
maximal normal subgroup of	C₂×C₃₇×SD₁₆⋊C₈, C₃₇×#17849, C₈×AGL(1,ℤ/8)×C₃₇, …
maximal quotient of	C₃₇×#8200, C₃₇×#8201, C₃₇×#8285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 310);