#82530

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×#41 ⋊ C₂
Sylow	C₃ ⋊ #3111

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×#41	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5238	C₂
hypercenter	C₄×#41	D₃
hypocenter	C₃	#3111
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

392 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3111, C₂×#82530, #11663403, …
maximal normal subgroup of	D₃×#3111, C₂×#82530, #11663403, …
maximal quotient of	#10795261, #10798203, #10798218, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15106);