#27085863

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3500 ⋊ C₂
Sylow	C₄₇ ⋊ #60651

Subgroups

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3500	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3500	D₄₇
hypocenter	C₄₇	#60651
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #3500

3138 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)(59,73)(64,77)(67,79)(72,82)(74,80)(76,83)(78,81);(52,55)(53,59)(54,64)(56,67)(57,66)(58,72)(60,74)(61,73)(62,76)(63,77)(65,78)(68,80)(69,79)(70,81)(71,82)(75,83);(48,49,50,51)(54,63)(58,71)(62,75)(64,77)(65,70)(72,82)(76,83)(78,81);(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)(52,54,56,65,57,63,69,70)(53,58,60,62,61,71,68,75)(55,64,67,78,66,77,79,81)(59,72,74,76,73,82,80,83);(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)(52,53,57,61)(54,62,63,75)(55,59,66,73)(56,68,69,60)(58,70,71,65)(64,76,77,83)(67,80,79,74)(72,81,82,78)

Polycyclic presentation:

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

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, 1039153);