#79450

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₈×D₄ ⋊ C₂
Sylow	C₃ ⋊ #3301

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₈×D₄	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₈×D₄	D₃
hypocenter	C₃	#3301
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

398 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)(6,11)(8,10);(2,3)(12,15,16,23)(13,19,20,27)(14,21,17,25)(18,26,24,22);(4,6)(5,8)(7,11)(9,10)(12,14,16,17)(13,18,20,24)(15,21,23,25)(19,26,27,22);(2,3)(4,5,7,9)(6,10,11,8)(12,13,16,20)(14,18,17,24)(15,22,23,26)(19,25,27,21)

Polycyclic presentation:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3301, C₂×#79450, #11671674, …
maximal normal subgroup of	D₃×#3301, C₂×#79450, #11671674, …
maximal quotient of	#10810632, #10811440, #10812386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12026);