#73667

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₄⋊C₄ ⋊ C₂
Sylow	C₃ ⋊ #2374

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	C₂²×D₁₂	C₂²
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#2374
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

552 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)(10,15)(12,14)(16,19,20,18)(17,21,22,23);(2,3)(9,13)(12,14)(16,18,20,19)(17,21,22,23);(4,6,7,5)(8,10)(9,12)(11,15)(13,14);(4,5,7,6)(8,9)(10,14)(11,13)(12,15)(16,17)(18,23)(19,21)(20,22)

Polycyclic presentation:

2^7,3|a^2=g,b^2=g,c^2=ef,d^2=ef,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	D₃×#2374, C₂×#73667, #10954910, …
maximal normal subgroup of	D₃×#2374, C₂×#73667, #10954910, …
maximal quotient of	#10716650, #10716742, #10716744, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 6243);