C₂×#72221835 #72244061

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product

C₂×#72221835

Splittings

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

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₃×C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle	C₃²×H(3,𝔽₃)	C₂²×C₃²
Frattini	C₃³	C₂×C₃²×C₃⋊D₃
Fitting	C₂×#14552273	C₂
socle	C₂×C₃³	C₃²×C₃⋊D₃
prime radical	C₂×C₃×C₃⋊#272	C₃²
hypercenter	C₂×C₃×C₉	C₃⋊((C₃⋊D₃)⋊C₃)
hypocenter	C₃×H(3,𝔽₃)	C₂²×3₋¹⁺²
LCS	G ▷ C₃²×H(3,𝔽₃) ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×C₉

2545 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 2545 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 972

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72221872
maximal normal subgroup of	C₂×#72221872
maximal quotient of	C₂×#72221836

GAP

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

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