#27079953

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄₇×#427 ⋊ C₂
Sylow	C₄₇ ⋊ #60108

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂²×#10625350	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#60108
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(52,56,58,57)(53,61,62,71)(54,65,67,66)(55,69,68,70)(59,63,73,77)(60,75,74,81)(64,79,78,80)(72,76,82,83);(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)(55,68)(60,74)(64,78)(69,70)(72,82)(75,81)(76,83)(79,80);(52,55)(53,60)(54,64)(56,69)(57,70)(58,68)(59,72)(61,75)(62,74)(63,76)(65,79)(66,80)(67,78)(71,81)(73,82)(77,83);(52,54,57,66,58,67,56,65)(53,59,61,63,62,73,71,77)(55,64,70,80,68,78,69,79)(60,72,75,76,74,82,81,83);(48,49,50,51)(52,53)(54,63)(55,60)(56,61)(57,71)(58,62)(59,66)(64,76)(65,73)(67,77)(68,74)(69,75)(70,81)(72,80)(78,83)(79,82)

Polycyclic presentation:

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

Conjugacy classes & character table

1376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033243);