#85834

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×#41 ⋊ C₂
Sylow	C₃ ⋊ #3428

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₃
commutator	C₂×C₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃
Fitting	C₃×C₄×#41	C₂
socle, cosocle	C₂²×C₃	C₂⁵
prime radical	C₂×D₃×#41	C₂
hypercenter	C₄×#41	D₃
hypocenter	C₃	#3428
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 23 points:

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

Polycyclic presentation:

2^7,3|a^2=g,e^2=f,b^a=bf,d^a=df,c^b=cf,h^d=h^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10792914, #10792917, #10794732, …
maximal normal subgroup of	#10792914, #10792917, #10794732, …
maximal quotient of	#10953399, #10953414, #10953416, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18410);