#73481

order	384
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₂²⋊C₄ ⋊ C₂
Sylow	C₃ ⋊ #2398

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁴
Frattini	C₂³	C₂³×D₃
Fitting	C₃×C₄×C₂²⋊C₄	C₂
prime radical	C₂²×#67	C₂²
hypercenter	C₄×C₂²⋊C₄	D₃
hypocenter	C₃	#2398
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

568 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);(5,7)(9,13)(10,15)(17,20)(21,23);(4,5)(6,7)(8,10,11,15)(9,12,13,14);(2,3)(8,9,11,13)(10,14,15,12)(16,17)(18,21)(19,20)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10803099, #10803100, #10803108, …
maximal normal subgroup of	#10803099, #10803100, #10803108, …
maximal quotient of	#10716884, #10716886, #10716888, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 6057);