#27092155

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3271 ⋊ C₂
Sylow	C₄₇ ⋊ #61078

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3271	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3271	D₄₇
hypocenter	C₄₇	#61078
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3271

2694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 3008

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)(53,62)(54,66)(60,70)(61,78)(64,71)(65,82)(68,81)(76,79);(52,56,58,69)(53,61,62,78)(54,65,66,82)(55,67,57,72)(59,74,75,83)(60,76,70,79)(63,80,73,77)(64,81,71,68);(49,51)(52,55,58,57)(53,60,62,70)(54,64,66,71)(56,67,69,72)(59,73,75,63)(61,76,78,79)(65,81,82,68)(74,77,83,80);(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)(52,54)(53,59)(55,64)(56,68)(57,71)(58,66)(60,73)(61,77)(62,75)(63,70)(65,67)(69,81)(72,82)(74,79)(76,83)(78,80);(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)(52,53,58,62)(54,63,66,73)(55,60,57,70)(56,61,69,78)(59,71,75,64)(65,80,82,77)(67,76,72,79)(68,83,81,74)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045445);