#27092473

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3288 ⋊ C₂
Sylow	C₄₇ ⋊ #61100

Subgroups

	subgroup	G/H
center	C₂	D₄²×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3288	C₂
socle	C₂×C₄₇	C₂×D₄²
hypercenter	#3288	D₄₇
hypocenter	C₄₇	#61100
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3288

2616 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

650 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045763);