#27083356

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄₇×#578 ⋊ C₂
Sylow	C₄₇ ⋊ #60433

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#578	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	C₂×#578	D₄₇
hypocenter	C₄₇	#60433
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#578

2678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 83 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)(55,66)(60,72)(63,77)(67,79)(73,82)(74,80)(76,83)(78,81);(52,55)(53,60)(54,63)(56,66)(57,67)(58,72)(59,73)(61,74)(62,76)(64,77)(65,78)(68,79)(69,80)(70,81)(71,82)(75,83);(52,54,56,64)(53,59,58,71)(55,63,66,77)(57,70,68,65)(60,73,72,82)(61,75,69,62)(67,81,79,78)(74,83,80,76);(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)(53,58)(59,71)(60,72)(61,69)(62,75)(73,82)(74,80)(76,83);(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)(54,62)(55,60)(56,58)(57,69)(59,65)(61,68)(63,76)(64,75)(66,72)(67,80)(70,71)(73,78)(74,79)(77,83)(81,82)

Polycyclic presentation:

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

Conjugacy classes & character table

1073 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036646);