#27106881

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3119 ⋊ C₂
Sylow	C₄₇ ⋊ #61861

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3119	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3119	D₄₇
hypocenter	C₄₇	#61861
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3119

1670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 6016

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

785 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1060171);