#27113362

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3303 ⋊ C₂
Sylow	C₄₇ ⋊ #62171

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3303	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3303	D₄₇
hypocenter	C₄₇	#62171
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3303

1708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 79 points:

(1,2)(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)(48,51,52,60)(49,54,55,59)(50,58,57,63)(53,61,56,62)(64,65)(66,71)(67,72)(68,70)(69,73)(74,76)(75,78)(77,79);(50,57)(53,56)(58,63)(61,62)(64,67,68,73)(65,72,70,69)(66,74,75,79)(71,76,78,77);(64,66)(65,71)(67,74)(68,75)(69,77)(70,78)(72,76)(73,79);(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)(48,50)(49,53)(51,58)(52,57)(54,61)(55,56)(59,62)(60,63)(65,70)(67,73)(71,78)(74,79);(48,49,52,55)(50,56,57,53)(51,59,60,54)(58,61,63,62)(64,65,68,70)(66,74,75,79)(67,72,73,69)(71,77,78,76)

Polycyclic presentation:

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

Conjugacy classes & character table

854 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1066652);