C₂×#72221872 #72244223

order	17496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#72221872

Splittings

Fitting	C₂×#14552273 ⋊ C₂²
Sylow	#14552273 ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#72221871
commutator, cosocle, hypocenter	#10624555	C₂³×C₃
Frattini	C₃³	C₂×C₃×D₃×C₃⋊D₃
Fitting	C₂×#14552273	C₂²
socle	C₂×C₃³	C₃×D₃×C₃⋊D₃
prime radical	C₂×#72221854	C₃
LCS	G ▷ #10624555
UCS	C₁ ◁ C₂×C₃

3381 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 3381 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 47
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 47 points:

(2,6,7,4,5,9)(3,8)(10,11)(12,15)(14,16)(19,25,41,22,35,23,29,20,31)(21,43,39,37,33,44,27,42,40,36,34,38,28,32,30,26,24,45);(1,7,4,3,5,6,8,2,9)(11,17,15,13,14,18)(12,16)(21,26)(24,32)(27,37)(28,36)(30,38)(33,43)(34,42)(39,45)(40,44);(19,21,26)(20,24,32)(22,28,36)(23,30,38)(25,34,42)(27,37,29)(31,40,44)(33,43,35)(39,45,41)(46,47)

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72221873

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,6,7,4,5,9)(3,8)(10,11)(12,15)(14,16)(19,25,41,22,35,23,29,20,31)(21,43,39,37,33,44,27,42,40,36,34,38,28,32,30,26,24,45), (1,7,4,3,5,6,8,2,9)(11,17,15,13,14,18)(12,16)(21,26)(24,32)(27,37)(28,36)(30,38)(33,43)(34,42)(39,45)(40,44), (19,21,26)(20,24,32)(22,28,36)(23,30,38)(25,34,42)(27,37,29)(31,40,44)(33,43,35)(39,45,41)(46,47));