#27092211

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#2932 ⋊ C₂
Sylow	C₄₇ ⋊ #61082

Subgroups

	subgroup	G/H
center	C₂²	C₂³×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#2932	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#2932	D₄₇
hypocenter	C₄₇	#61082
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2932

2810 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(52,54,56,61)(53,59,60,66)(55,62,63,67)(57,64,58,65);(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)(53,58)(55,63)(57,60)(59,65)(62,67)(64,66);(49,51)(54,61)(59,66)(62,67)(64,65);(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)(52,53)(54,59)(55,58)(56,60)(57,63)(61,66)(62,65)(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)(48,49)(50,51)(53,57)(54,61)(55,63)(58,60)(59,65)(64,66)

Polycyclic presentation:

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

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045501);