#73779

order	384
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₄⋊C₄ ⋊ C₂
Sylow	C₃ ⋊ #2401

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁴
Frattini	C₂³	C₂³×D₃
Fitting	C₃×C₄×C₄⋊C₄	C₂
prime radical	C₂³×D₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#2401
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

432 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)(20,22,23,26)(21,25,24,27);(2,3)(5,9)(8,11)(14,18)(16,19);(4,6,7,10)(5,8,9,11)(12,14)(13,16)(15,18)(17,19)(21,24)(25,27);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,16,18,19)(20,21)(22,25)(23,24)(26,27)

Polycyclic presentation:

2^7,3|a^2=f,b^2=fg,d^2=e,b^a=be,c^a=cfg,h^a=h^2,c^b=cg,h^c=h^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10803443, #10803446, #10803450, …
maximal normal subgroup of	#10803443, #10803446, #10803450, …
maximal quotient of	#10716886, #10716889, #10716891, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 6355);