#73093

order	384
isoclinism	isoclinic class of #73085
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	GL(2,ℤ/4)
commutator, cosocle	C₄×SL(2,𝔽₃)	C₂²
Frattini	C₂×C₄	C₂×S₄
Fitting	#557	D₃
prime radical	C₂×SL(2,𝔽₃)	D₄
hypercenter	C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	Q₁₆
LCS	G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,23,27,18,28,31,22,25,29,19,20,32)(21,26,30,24);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,19)(20,26,28,24)(21,25,30,23)(27,32,29,31)

Polycyclic presentation:

2^4,3,2^3|a^2=dh,b^2=dh,c^2=d,f^2=h,g^2=h,b^a=bc,c^a=cd,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#73093, #11711962, #11711983, …
maximal normal subgroup of	C₂×#73093, #11711962, #11711983, …
maximal quotient of	#11711055, #11711096, #11711118, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5669);