#82210

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2924

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×#117	C₂
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#117	D₃
hypocenter	C₃	#2924
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10691499, #10691503, #10691513, …
maximal normal subgroup of	#10691499, #10691503, #10691513, …
maximal quotient of	#10792685, #10792748, #10793099, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14786);