#27123183

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3531 ⋊ C₂
Sylow	C₄₇ ⋊ #62694

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₄₇×#3531	C₂
hypercenter	#3531	D₄₇
hypocenter	C₄₇	#62694
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #3531

6640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 6 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)(64,67,68,75)(65,71,70,77)(66,74,73,79)(69,76,72,78);(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)(48,51,52,58)(49,55,54,61)(50,57,56,62)(53,60,59,63);(50,56)(53,59)(57,62)(60,63)(65,70)(69,72)(71,77)(76,78);(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)(49,54)(50,56)(55,61)(57,62)(66,73)(69,72)(74,79)(76,78);(48,50)(49,53)(51,57)(52,56)(54,59)(55,60)(58,62)(61,63)(64,66)(65,69)(67,74)(68,73)(70,72)(71,76)(75,79)(77,78);(48,49)(50,53)(51,55)(52,54)(56,59)(57,60)(58,61)(62,63)(64,65)(66,72)(67,71)(68,70)(69,73)(74,78)(75,77)(76,79)

Polycyclic presentation:

2^8,47|e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,i^c=i^46,i^e=i^46,i^f=i^46

Conjugacy classes & character table

950 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076473);