#79923

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₈×Q₈ ⋊ C₂
Sylow	C₃ ⋊ #2986

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₈×Q₈	C₂
prime radical	C₂²×D₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₈×Q₈	D₃
hypocenter	C₃	#2986
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2)(5,11)(9,12)(10,19)(17,18)(22,28)(25,31)(29,34)(32,35);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(22,28)(25,31)(29,34)(32,35);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35);(2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10766362, #10766365, #10766399, …
maximal normal subgroup of	#10766362, #10766365, #10766399, …
maximal quotient of	#10793945, #10793952, #10794239, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12499);