D₁₉⋊C₃×ΓL(1,𝔽₈)⋊D₇ #45692014

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

Aliases

direct product	D₁₉⋊C₃×ΓL(1,𝔽₈)⋊D₇
unambiguous semidirect product	(ΓL(1,𝔽₈)×D₁₉⋊C₃)⋊D₇

Splittings

Fitting, socle, derived, Hall	C₇²×C₁₉ ⋊ C₂²×C₃²
Hall	C₇×ΓL(1,𝔽₈)×C₁₉⋊C₃ ⋊ C₂²
Hall	D₁₉×C₇⋊D₇ ⋊ C₃²
Sylow	C₇² ⋊ C₂×C₃×D₁₉⋊C₃
Sylow	C₁₉ ⋊ C₂×C₃×ΓL(1,𝔽₈)⋊D₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₁₉	C₂²×C₃²
LCS	G ▷ C₇²×C₁₉
UCS	C₁

280 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 399

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

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

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, 812);