#80215

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #3116

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#452	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×#41	C₂²
hypercenter	#452	D₃
hypocenter	C₃	#3116
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

386 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3116, C₂×#80215, #11663394, …
maximal normal subgroup of	D₃×#3116, C₂×#80215, #11663394, …
maximal quotient of	#10795263, #10798196, #10798204, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12791);