#83597

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #3352

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#552	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×D₄	D₄
hypercenter	#552	D₃
hypocenter	C₃	#3352
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #552

336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2,3)(6,11)(9,10)(13,19)(17,25)(18,20)(24,26);(4,7,8,5)(6,10,11,9)(13,18)(14,21)(16,23)(17,24)(19,20)(25,26);(4,6)(5,9)(7,10)(8,11)(12,14)(13,17)(15,21)(16,22)(18,26)(19,25)(20,24)(23,27);(2,3)(4,5,8,7)(6,10,11,9)(12,13,15,19)(14,17,21,25)(16,18,23,20)(22,26,27,24)

Polycyclic presentation:

2^7,3|a^2=e,c^2=eg,f^2=g,b^a=beg,c^a=cf,d^a=dg,f^a=fg,h^a=h^2,c^b=cg,d^b=deg,f^c=fg

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3352, C₂×#83597, #11673941, …
maximal normal subgroup of	D₃×#3352, C₂×#83597, #11673941, …
maximal quotient of	#10811589, #10811595, #10812965, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16173);