#73768

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₄⋊C₄ ⋊ C₂
Sylow	C₃ ⋊ #2345

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×C₄⋊C₄	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂³×D₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#2345
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

548 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);(2,3)(9,13)(12,15)(17,20)(21,23);(8,10,11,14)(9,12,13,15)(17,20)(18,22);(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,21)(19,20)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10812278, #10812279, #10812289, …
maximal normal subgroup of	#10812278, #10812279, #10812289, …
maximal quotient of	C₄⋊C₄×(C₂×D₃)⋊C₄, #10716621, #10716676, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 6344);