#27122816

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3584 ⋊ C₂
Sylow	C₄₇ ⋊ #62636

Subgroups

	subgroup	G/H
center	C₂	C₂³×#10625334
commutator, cosocle	C₈×C₄₇	C₂⁵
Frattini	C₈	C₂⁴×D₄₇
Fitting	C₄₇×#3584	C₂
socle	C₂×C₄₇	C₂³×D₈
hypercenter	#3584	D₄₇
hypocenter	C₄₇	#62636
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ #3584

2078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 1504

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

Polycyclic presentation:

2^8,47|f^2=gh,g^2=h,b^a=bf,e^a=eh,f^a=fg,g^a=gh,f^b=fg,g^b=gh,i^b=i^46,d^c=dh

Conjugacy classes & character table

1070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076106);