C₂×#26039518 #37525002

order	22932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₂×#26039518

Splittings

Fitting, socle	C₂×C₇²×C₁₃ ⋊ C₂×C₃²
derived, Hall	C₇²×C₁₃ ⋊ C₂²×C₃²
Hall	#19153211 ⋊ C₂²
Hall	C₂×C₇⋊D₉₁ ⋊ C₃²
Sylow	C₇² ⋊ C₂×C₃×D₁₃⋊C₃
Sylow	C₁₃ ⋊ C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#26039518
commutator, cosocle, hypocenter	C₇²×C₁₃	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₃	C₂×C₃²
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂

280 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 29
transitive degree	μ_t(G) = 546

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22932, 947);