#27124055

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3529 ⋊ C₂
Sylow	C₄₇ ⋊ #62755

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₄₇×#3529	C₂
hypercenter	#3529	D₄₇
hypocenter	C₄₇	#62755
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #3529

6644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 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)(48,51,52,59)(49,55,54,61)(50,58,57,63)(53,60,56,62)(64,68,69,67)(65,71,72,76)(66,74,75,77)(70,79,73,78);(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)(65,72)(70,73)(71,76)(78,79);(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)(53,56)(55,61)(60,62)(64,67,69,68)(65,71,72,76)(66,74,75,77)(70,78,73,79);(50,57)(53,56)(58,63)(60,62)(64,67,69,68)(65,71,72,76)(66,74,75,77)(70,78,73,79);(48,50)(49,53)(51,58)(52,57)(54,56)(55,60)(59,63)(61,62)(64,66)(65,70)(67,77)(68,74)(69,75)(71,79)(72,73)(76,78);(48,49)(50,56)(51,55)(52,54)(53,57)(58,62)(59,61)(60,63)(64,65)(66,73)(67,76)(68,71)(69,72)(70,75)(74,78)(77,79)

Polycyclic presentation:

2^8,47|c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh,i^d=i^46,i^e=i^46

Conjugacy classes & character table

950 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1077345);