#79722

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×Q₁₆ ⋊ C₂
Sylow	C₃ ⋊ #3104

Subgroups

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×Q₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#3104
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

384 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);(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(2,3)(5,8)(9,11);(4,5)(6,9)(7,8)(10,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)

Polycyclic presentation:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10765077, #10765079, #10765110, …
maximal normal subgroup of	#10765077, #10765079, #10765110, …
maximal quotient of	#10795197, #10798177, #10798179, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12298);