#27083024

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3028 ⋊ C₂
Sylow	C₄₇ ⋊ #60398

Subgroups

	subgroup	G/H
center	C₂	C₂×#120×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₄₇×#3028	C₂
socle	C₂×C₄₇	C₂²×#120
hypercenter	#3028	D₄₇
hypocenter	C₄₇	#60398
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3028

4836 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)(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)(48,52,53,63)(49,57,56,69)(50,59,60,73)(51,61,62,74)(54,66,65,76)(55,68,67,77)(58,71,72,79)(64,75,70,78);(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)(49,56)(54,65)(55,67)(57,69)(64,70)(66,76)(68,77)(75,78);(48,51)(49,55)(50,58)(52,61)(53,62)(54,64)(56,67)(57,68)(59,71)(60,72)(63,74)(65,70)(66,75)(69,77)(73,79)(76,78);(48,50)(49,54)(51,58)(52,59)(53,60)(55,64)(56,65)(57,66)(61,71)(62,72)(63,73)(67,70)(68,75)(69,76)(74,79)(77,78);(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)(48,49)(50,54)(51,55)(52,57)(53,56)(58,70)(59,66)(60,65)(61,68)(62,67)(63,69)(64,72)(71,78)(73,76)(74,77)(75,79)

Polycyclic presentation:

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

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036314);