#79004

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 ⋊ #2935

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄⋊C₈	C₂
prime radical	#5477	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#2935
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

492 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)(8,11)(9,15)(10,17)(12,20)(13,19)(14,21)(16,22)(18,23);(2,3)(8,10,12,18,13,9,14,16)(11,15,20,22,19,17,21,23);(2,3)(5,7)(11,19)(15,17)(20,21)(22,23);(4,5)(6,7)(8,9,12,16,13,10,14,18)(11,15,20,22,19,17,21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10689662, #10689666, #10689676, …
maximal normal subgroup of	#10689662, #10689666, #10689676, …
maximal quotient of	#10766944, #10766952, #10766963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11580);