#27080614

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#2916 ⋊ C₂
Sylow	C₄₇ ⋊ #60183

Subgroups

	subgroup	G/H
center	C₄	C₂²×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#2916	C₂
socle	C₂×C₄₇	C₂²×C₄×D₄
prime radical	#20258162	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#2916	D₄₇
hypocenter	C₄₇	#60183
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #2916

2612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 5 generators on 79 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)(52,65)(56,61)(62,75)(66,73)(67,74)(70,71)(76,79)(77,78);(48,52)(49,56)(50,61)(51,62)(53,66)(54,65)(55,67)(57,70)(58,71)(59,73)(60,74)(63,76)(64,75)(68,77)(69,78)(72,79);(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)(48,51,53,63,54,64,59,72)(49,55,58,69,50,60,57,68)(52,62,66,76,65,75,73,79)(56,67,71,78,61,74,70,77);(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)(49,54)(51,60)(52,61)(53,57)(55,64)(56,65)(58,59)(62,74)(63,68)(66,70)(67,75)(69,72)(71,73)(76,77)(78,79);(48,49,53,58,54,50,59,57)(51,55,63,69,64,60,72,68)(52,56,66,71,65,61,73,70)(62,67,76,78,75,74,79,77)

Polycyclic presentation:

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

Conjugacy classes & character table

1100 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033904);