C₂×C₃⋊#16838456 #37525534

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

Aliases

direct product	C₂×C₃⋊#16838456
unambiguous semidirect product	(C₂×C₃)⋊#16838456

Splittings

Fitting, socle	C₂×C₃×C₇²×C₁₃ ⋊ C₂×C₃
derived	C₃×C₇²×C₁₃ ⋊ C₂²×C₃
Hall	C₃×#14274263 ⋊ C₂²
Hall	C₇²×C₁₃ ⋊ C₂×C₃×D₃
Sylow	C₇² ⋊ C₂×C₃⋊(D₁₃⋊C₃)
Sylow	C₁₃ ⋊ C₂×C₃⋊#64423

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#16838456
commutator, cosocle, hypocenter	C₃×C₇²×C₁₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₃×C₇²×C₁₃
UCS	C₁ ◁ C₂

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 1092

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22932, 1479);