#82435

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2982

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator	C₂×C₃	C₂³×C₈
Fitting	C₂×C₃×C₄×Q₈	C₂
socle	C₂²×C₃	C₂³×C₄
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄×Q₈	D₃
hypocenter	C₃	#2982
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10689053, #10689056, #10689835, …
maximal normal subgroup of	#10689053, #10689056, #10689835, …
maximal quotient of	#10793918, #10794227, #10804598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15011);