#73452

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₂²⋊C₄ ⋊ C₂
Sylow	C₃ ⋊ #2337

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×C₂²⋊C₄	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂²×#67	C₂²
hypercenter	C₄×C₂²⋊C₄	D₃
hypocenter	C₃	#2337
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

540 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)(9,12)(13,14)(16,18,19,22)(17,21,20,23);(2,3)(8,10,11,15)(9,13,12,14)(17,20)(21,23);(4,6,7,5)(9,12)(10,15);(4,5,7,6)(8,9,11,12)(10,14,15,13)(16,17)(18,21)(19,20)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#2337, C₂×#73452, #10954111, …
maximal normal subgroup of	D₃×#2337, C₂×#73452, #10954111, …
maximal quotient of	#10716459, #10716516, #10716524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 6028);