#45692005

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

Aliases

No aliases for this group.

Splittings

Fitting, socle, Hall	C₇²×C₁₉ ⋊ C₂×C₃×D₃
derived	C₁₉×#3862 ⋊ C₂²×C₃
Hall	#22552728 ⋊ C₂²
Sylow	C₇² ⋊ C₂×D₃×C₁₉⋊C₃
Sylow	C₁₉ ⋊ (#64418⋊C₂)⋊C₃

Subgroups

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

152 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 33
transitive degree	μ_t(G) = 266

A minimal faithful permutation representation — 2 generators on 33 points:

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

Polycyclic presentation:

2^2,3^2,19,7^2|d^a=d^2,f^a=g^5,g^a=f^3,d^b=d^2,f^b=g^2,g^b=f^4,e^c=e^7,f^c=f^2,g^c=g^2,f^d=f^4,g^d=g^2

Conjugacy classes & character table

139 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 803);