#27091716

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3266 ⋊ C₂
Sylow	C₄₇ ⋊ #61057

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3266	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3266	D₄₇
hypocenter	C₄₇	#61057
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3266

2696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 83 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(52,56,58,57)(53,61,63,62)(54,66,67,71)(55,69,70,72)(59,64,74,78)(60,76,77,79)(65,81,82,68)(73,80,83,75);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(52,55)(53,60)(54,65)(56,69)(57,72)(58,70)(59,73)(61,76)(62,79)(63,77)(64,80)(66,81)(67,82)(68,71)(74,83)(75,78);(49,51);(52,54,58,67)(53,59,63,74)(55,68,70,81)(56,66,57,71)(60,75,77,80)(61,64,62,78)(65,72,82,69)(73,79,83,76);(48,49)(50,51)(52,53,58,63)(54,64,67,78)(55,60,70,77)(56,62,57,61)(59,71,74,66)(65,80,82,75)(68,83,81,73)(69,79,72,76)

Polycyclic presentation:

2^8,47|a^2=h,b^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,f^b=fh,f^d=fh,i^e=i^46

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045006);