#78604

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₂²⋊C₈ ⋊ C₂
Sylow	C₃ ⋊ #2976

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₂²⋊C₈	C₂
prime radical	#5791	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₃
hypocenter	C₃	#2976
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

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

Polycyclic presentation:

2^7,3|a^2=f,f^2=g,b^a=be,d^a=de,c^b=ce,d^b=dg,h^b=h^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10689290, #10689294, #10689297, …
maximal normal subgroup of	#10689290, #10689294, #10689297, …
maximal quotient of	#10793881, #10793901, #10793959, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11180);