#80822

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#468 ⋊ C₂
Sylow	C₃ ⋊ #3313

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#468	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5598	C₂
hypercenter	#468	D₃
hypocenter	C₃	#3313
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

380 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)(6,11)(8,10)(12,14,17,16)(13,18,20,25)(15,21,24,23)(19,26,27,22);(2,3)(12,15,17,24)(13,19,20,27)(14,21,16,23)(18,26,25,22);(4,6)(5,8)(7,11)(9,10)(12,14,17,16)(13,18,20,25)(15,23,24,21)(19,22,27,26);(4,5,7,9)(6,10,11,8)(12,13)(14,18)(15,22)(16,25)(17,20)(19,21)(23,27)(24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3313, C₂×#80822, #11671950, …
maximal normal subgroup of	D₃×#3313, C₂×#80822, #11671950, …
maximal quotient of	#10810767, #10811515, #10812316, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 13398);