#27091571

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3284 ⋊ C₂
Sylow	C₄₇ ⋊ D₄×#135

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3284	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3284	D₄₇
hypocenter	C₄₇	D₄×#135
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3284

2822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 3008

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)(52,55,57,63)(53,59,60,66)(54,61,56,62)(58,65,64,67);(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)(55,61)(56,57)(58,64)(62,63)(65,67);(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)(49,51);(52,53)(54,58)(55,59)(56,64)(57,60)(61,65)(62,67)(63,66);(48,49)(50,51)(53,58)(54,56)(59,65)(60,64)(61,62)(66,67)

Polycyclic presentation:

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

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1044861);