#79016

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₄⋊C₈ ⋊ C₂
Sylow	C₃ ⋊ #2992

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄⋊C₈	C₂
prime radical	#930	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	#2992
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

356 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)(4,6,7,10)(5,9,8,11)(14,20)(17,23)(21,26)(24,27);(5,8)(9,11);(2,3)(4,5)(6,9)(7,8)(10,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(5,8)(6,10)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10689304, #10689308, #10689311, …
maximal normal subgroup of	#10689304, #10689308, #10689311, …
maximal quotient of	#10793984, #10794051, #10794129, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11592);