C₂×(C₁₉⋊C₃)⋊#64425 #45692191

order	33516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃×C₇

Aliases

direct product	C₂×(C₁₉⋊C₃)⋊#64425
unambiguous semidirect product	(C₂×C₁₉⋊C₃)⋊#64425

Splittings

Fitting, socle	C₂×C₇²×C₁₉ ⋊ C₂×C₃²
derived, Hall	C₇²×C₁₉ ⋊ C₂²×C₃²
Hall	C₁₉⋊C₃×#3862 ⋊ C₂²
Hall	C₂×C₇⋊D₁₃₃ ⋊ C₃²
Sylow	C₇² ⋊ C₂×C₃×D₁₉⋊C₃
Sylow	C₁₉ ⋊ C₂×C₃×#64425

Subgroups

	subgroup	G/H
center, hypercenter	C₂	(C₁₉⋊C₃)⋊#64425
commutator, cosocle, hypocenter	C₇²×C₁₉	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₉	C₂×C₃²
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₂

280 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 35
transitive degree	μ_t(G) = 798

A minimal faithful permutation representation — 2 generators on 35 points:

(4,7,8,5,6,9)(11,15,14,12,13,16)(17,18)(19,21)(20,28)(22,24)(23,29)(25,27)(26,34)(30,31)(32,33);(1,2)(3,4,6,9,8,7,5)(10,11,14,15,13,16,12)(18,20,23)(19,22,24)(21,26,27)(25,30,32)(28,31,35)(29,34,33)

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 989);