#27083239

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3042 ⋊ C₂
Sylow	C₄₇ ⋊ #60423

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3042	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3042	D₄₇
hypocenter	C₄₇	#60423
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3042

2890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 5 generators on 67 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)(53,59)(57,65)(58,60)(64,66);(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)(54,61)(57,65)(62,67)(64,66);(53,58)(55,63)(57,64)(59,60)(62,67)(65,66);(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)(52,54)(53,57)(55,62)(56,61)(58,64)(59,65)(60,66)(63,67);(49,51)(52,53)(54,57)(55,58)(56,59)(60,63)(61,65)(62,64)(66,67)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036529);