C₂×D₃×#14274262 #37525525

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

Aliases

direct product	C₂×D₃×#14274262
unambiguous semidirect product	C₃⋊(C₂²×#14274262)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×#14274262)

Splittings

Fitting, socle	C₂×C₃×C₇²×C₁₃ ⋊ C₂×C₃
derived	C₃×C₇²×C₁₃ ⋊ C₂²×C₃
Hall	C₃×#14274262 ⋊ C₂²
Hall	C₇²×C₁₃ ⋊ C₂×C₃×D₃
Sylow	C₇² ⋊ C₂×D₃×C₁₃⋊C₃
Sylow	C₁₃ ⋊ C₂×D₃×#3861

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×#14274262
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

440 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 3822

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

1290 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22932, 1470);