#27083339

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3046 ⋊ C₂
Sylow	C₄₇ ⋊ #60432

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3046	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3046	D₄₇
hypocenter	C₄₇	#60432
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3046

2646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 83 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)(55,66)(59,73)(63,77)(67,79)(71,82)(74,80)(76,83)(78,81);(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)(52,55)(53,59)(54,63)(56,67)(57,66)(58,71)(60,74)(61,73)(62,76)(64,78)(65,77)(68,80)(69,81)(70,79)(72,82)(75,83);(52,54,57,65)(53,58,61,72)(55,63,66,77)(56,69,70,64)(59,71,73,82)(60,75,68,62)(67,81,79,78)(74,83,80,76);(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)(48,49,50,51);(49,51)(52,53,57,61)(54,62,65,75)(55,59,66,73)(56,68,70,60)(58,69,72,64)(63,76,77,83)(67,80,79,74)(71,81,82,78)

Polycyclic presentation:

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

Conjugacy classes & character table

797 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036629);