#68047

order	384
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₈² ⋊ C₂
Sylow	C₃ ⋊ #1450

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₃
Fitting	C₃×C₈²	C₂
socle	C₂²×C₃	C₂×C₄²
prime radical	C₂²×D₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	C₈²	D₃
hypocenter	C₃	#1450
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10704167, #10704170, #10704175, …
maximal normal subgroup of	#10704167, #10704170, #10704175, …
maximal quotient of	#10684542, #10684571, #10685615, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 623);