#69547

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1835

Subgroups

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#382	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#382	D₃
hypocenter	C₃	#1835
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2)(9,13)(12,15)(16,19,20,30,22,29,21,18)(17,24,25,23,27,28,26,31);(2,3)(8,10,11,14)(9,12,13,15)(16,18,21,29,22,30,20,19)(17,23,26,24,27,31,25,28);(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,28)(19,23)(20,26)(21,25)(22,27)(24,30)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682593, #10688409, #10688410, …
maximal normal subgroup of	#10682593, #10688409, #10688410, …
maximal quotient of	#10688850, #10688860, #10688871, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 2123);