#27080620

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#2956 ⋊ C₂
Sylow	C₄₇ ⋊ #60184

Subgroups

	subgroup	G/H
center	C₄	C₂²×D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#2956	C₂
socle	C₂×C₄₇	C₂²×C₄×D₄
prime radical	#20258161	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	#2956	D₄₇
hypocenter	C₄₇	#60184
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #2956

2604 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)(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)(52,65)(56,61)(62,75)(66,73)(68,74)(71,72)(76,77)(78,79);(48,52)(49,56)(50,61)(51,62)(53,66)(54,65)(55,68)(57,71)(58,72)(59,73)(60,74)(63,76)(64,75)(67,77)(69,78)(70,79);(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)(48,51,53,63,54,64,59,67)(49,55,58,70,50,60,57,69)(52,62,66,76,65,75,73,77)(56,68,72,79,61,74,71,78);(48,50)(49,54)(51,60)(52,61)(53,57)(55,64)(56,65)(58,59)(62,74)(63,69)(66,71)(67,70)(68,75)(72,73)(76,78)(77,79);(48,49,53,58,54,50,59,57)(51,60,63,69,64,55,67,70)(52,56,66,72,65,61,73,71)(62,74,76,78,75,68,77,79)

Polycyclic presentation:

2^8,47|a^2=g,c^2=g,f^2=h,g^2=h,b^a=bf,c^a=ch,f^a=fh,f^b=fh,i^c=i^46,e^d=eh,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, 1033910);