#79677

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×SD₁₆ ⋊ C₂
Sylow	C₃ ⋊ #3296

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×SD₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5554	C₂
hypercenter	C₄×SD₁₆	D₃
hypocenter	C₃	#3296
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3296, C₂×#79677, #11671674, …
maximal normal subgroup of	D₃×#3296, C₂×#79677, #11671674, …
maximal quotient of	#10810609, #10811417, #10812322, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12253);