C₂×#30660700 #45692117

order	33516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#30660700

Splittings

Fitting, socle	C₂×C₇²×C₁₉ ⋊ C₂×C₉
derived, Hall	C₇²×C₁₉ ⋊ C₂²×C₉
Hall	C₇×#11924781 ⋊ C₂²
Hall	C₂×C₇⋊D₁₃₃ ⋊ C₉
Sylow	C₇² ⋊ C₂×AGL(1,𝔽₁₉)
Sylow	C₁₉ ⋊ C₂×(C₇⋊C₉)⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#30660700
commutator, hypocenter	C₇²×C₁₉	C₂²×C₉
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₉	C₂×C₉
prime radical	C₂×(C₁₉⋊C₃)⋊(C₇⋊D₇)	C₃
cosocle	C₇²×C₁₉⋊C₃	C₂²×C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₂

160 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 37 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)(11,12)(13,14)(15,16)(18,30,33,28,26,31,21,23,32,19,34,29,27,22,25,24,20,35);(1,2)(3,4,6,9,8,7,5)(10,11,13,15,16,14,12)(17,18,22,23,32,34,33,21,28,25,31,27,24,29,30,35,20,26,19)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 915);