C₂×#72221846 #72244126

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

C₂×#72221846

Splittings

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

Subgroups

	subgroup	G/H
center	C₂×C₃	C₃×#4169⋊C₃
commutator, cosocle	C₃²×C₉	C₂²×C₃³
Frattini	C₃³	C₂×C₃³×D₃
Fitting	C₂×#14552273	C₂
socle	C₂×C₃³	C₃³×D₃
prime radical	C₂×#13090639	C₃
hypercenter	C₂×C₃²	#4169⋊C₃
hypocenter	C₃×C₉	C₂²×#737
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃²

2673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 324

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

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

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×#72221847

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

C2×#72221846 #72244126