C₂×C₃×#16838468 #37525450

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

Aliases

direct product

C₂×C₃×#16838468

Splittings

Fitting, socle	C₂×C₃×C₇²×C₁₃ ⋊ C₂×C₃
derived, Hall	C₇²×C₁₃ ⋊ C₂²×C₃²
Hall	C₃×C₇×#63270 ⋊ C₂²
Hall	C₂×C₇⋊D₉₁ ⋊ C₃²
Sylow	C₇² ⋊ C₂×C₃×D₁₃⋊C₃
Sylow	C₁₃ ⋊ C₂×C₃×ΓL(1,𝔽₈)⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#16838468
commutator, cosocle, hypocenter	C₇²×C₁₃	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂×C₃

280 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 32
transitive degree	μ_t(G) = 1638

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

720 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22932, 1395);