#79012

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₄⋊C₈ ⋊ C₂
Sylow	C₃ ⋊ #2906

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄⋊C₈	C₂
prime radical	C₂×#235	C₂²
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#2906
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

484 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)(10,16)(13,19)(17,22)(20,23);(2,3)(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23);(4,5,6,7);(5,7)(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23)

Polycyclic presentation:

2^7,3|a^2=f,b^2=e,f^2=g,b^a=be,d^c=dg,h^c=h^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10689486, #10689490, #10689500, …
maximal normal subgroup of	#10689486, #10689490, #10689500, …
maximal quotient of	#10764408, #10764413, #10764414, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11588);