#27088368

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3232 ⋊ C₂
Sylow	C₄₇ ⋊ #60846

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3232	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#20256682	C₂
hypercenter	#3232	D₄₇
hypocenter	C₄₇	#60846
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3232

1800 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)(64,67,69,68)(65,71,72,76)(66,74,75,77)(70,78,79,73);(49,55)(53,56)(54,63)(61,62);(48,51,52,60)(49,54,55,63)(50,57,58,59)(53,61,56,62)(64,66)(65,70)(67,74)(68,77)(69,75)(71,78)(72,79)(73,76);(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)(48,50,52,58)(49,53,55,56)(51,59,60,57)(54,62,63,61);(48,49)(50,56)(51,54)(52,55)(53,58)(57,62)(59,61)(60,63)(64,65)(66,73)(67,71)(68,76)(69,72)(70,74)(75,78)(77,79)

Polycyclic presentation:

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

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1041658);