#79726

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 ⋊ #3172

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	#895	C₂²
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#3172
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

364 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);(12,14,15,22,16,21,24,25)(13,17,18,20,19,26,23,27);(2,3)(5,8)(9,11)(14,21)(17,26)(20,27)(22,25);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,19)(14,20,21,27)(15,23,24,18)(17,25,26,22)

Polycyclic presentation:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10765779, #10765783, #10765799, …
maximal normal subgroup of	#10765779, #10765783, #10765799, …
maximal quotient of	#10795739, #10799429, #10799431, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12302);