C₂×#30660725 #45692102

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

Aliases

direct product

C₂×#30660725

Splittings

Fitting, socle	C₂×C₇²×C₁₉ ⋊ C₃×D₃
derived	C₁₉×#3862 ⋊ C₂²×C₃
Hall	#22552727 ⋊ C₂²
Hall	C₇²×C₁₉ ⋊ C₂×C₃×D₃
Sylow	C₇² ⋊ C₂×C₃⋊(D₁₉⋊C₃)
Sylow	C₁₉ ⋊ C₂×#64418⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#30660725
commutator, cosocle, hypocenter	C₁₉×#3862	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₉	C₃×D₃
LCS	G ▷ C₁₉×#3862
UCS	C₁ ◁ C₂

152 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 532

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

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 900);