#73668

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₄⋊C₄ ⋊ C₂
Sylow	C₃ ⋊ #2396

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	#2396
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10690275, #10690280, #10767467, …
maximal normal subgroup of	#10690275, #10690280, #10767467, …
maximal quotient of	#10716900, #10716914, #10716932, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 6244);