#67508

order	384
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1397

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₃
commutator	C₂×C₃	C₄×C₁₆
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×C₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#1397
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3)(4,5,6,7)(9,14)(15,24)(16,25)(17,26)(27,34)(28,35)(29,36)(37,39);(2,3)(8,9,10,15,11,16,18,27,12,17,19,28,21,29,30,37,13,14,20,24,22,25,31,34,23,26,32,35,33,36,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682516, #10682517, #10682520, …
maximal normal subgroup of	#10682516, #10682517, #10682520, …
maximal quotient of	#10682515, #10682518, #10682874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 84);