#27080471

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#2950 ⋊ C₂
Sylow	C₄₇ ⋊ #60169

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#2950	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂²×#10625354	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#2950	D₄₇
hypocenter	C₄₇	#60169
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #2950

2666 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)(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)(53,60)(58,72)(59,73)(61,68)(62,75)(71,82)(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)(55,66)(59,73)(63,77)(67,79)(71,82)(74,80)(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)(48,50,51,49)(52,55)(53,59)(54,63)(56,67)(57,66)(58,71)(60,73)(61,74)(62,76)(64,78)(65,77)(68,80)(69,81)(70,79)(72,82)(75,83);(52,54,57,65)(53,58,60,72)(55,63,66,77)(56,69,70,64)(59,71,73,82)(61,75,68,62)(67,81,79,78)(74,83,80,76);(48,49,51,50)(52,53)(54,62)(55,59)(56,68)(57,60)(58,64)(61,70)(63,76)(65,75)(66,73)(67,80)(69,72)(71,78)(74,79)(77,83)(81,82)

Polycyclic presentation:

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

Conjugacy classes & character table

1094 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033761);