#82525

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×#41 ⋊ C₂
Sylow	C₃ ⋊ #3035

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×#41	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₄×#41	D₃
hypocenter	C₃	#3035
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10690072, #10690074, #10690079, …
maximal normal subgroup of	#10690072, #10690074, #10690079, …
maximal quotient of	#10777609, #10777624, #10777635, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15101);