#27083541

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×AGL(1,ℤ/8)×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ #60454

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×AGL(1,ℤ/8)×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	C₂²×AGL(1,ℤ/8)	D₄₇
hypocenter	C₄₇	#60454
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×AGL(1,ℤ/8)

2870 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036831);