#83553

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×D₄×Q₈ ⋊ C₂
Sylow	C₃ ⋊ #3320

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×D₄×Q₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5565	C₂
hypercenter	D₄×Q₈	D₃
hypocenter	C₃	#3320
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ D₄×Q₈

388 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);(12,15,16,24)(13,19,20,27)(14,21,17,23)(18,26,25,22);(4,6)(5,8)(7,11)(9,10)(12,14,16,17)(13,18,20,25)(15,23,24,21)(19,22,27,26);(2,3)(4,5,7,9)(6,10,11,8)(12,13,16,20)(14,18,17,25)(15,22,24,26)(19,23,27,21)

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,c^b=cg,d^b=deg,f^c=fg

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3320, C₂×#83553, #11671957, …
maximal normal subgroup of	D₃×#3320, C₂×#83553, #11671957, …
maximal quotient of	#10810757, #10811499, #10812291, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16129);