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#997

order	96
automorphism	Aut(G) = (#1020⋊C₃)⋊C₂ (order 576) · Inn(G) = C₂²⋊A₄ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(5,𝔽₂) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	C₂²⋊A₄
commutator, Fitting, cosocle, hypocenter	H(5,𝔽₂)	C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 3 generators on 8 points:

(2,3,5)(6,7,8);(2,6)(3,7);(1,5)(2,3)(4,8)(6,7)

Polycyclic presentation:

3,2^5|b^a=c,c^a=bc,d^a=e,e^a=de,e^b=ef,d^c=df

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	1	6	6	6	6	6	16	16	16	16
order	1	2	2	2	2	4	4	3	3	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	3	3	-1	-1	-1	-1	0	0	0	0
χ₅	3	3	-1	3	-1	-1	-1	0	0	0	0
χ₆	3	3	-1	-1	3	-1	-1	0	0	0	0
χ₇	3	3	-1	-1	-1	3	-1	0	0	0	0
χ₈	3	3	-1	-1	-1	-1	3	0	0	0	0
χ₉	4	-4	0	0	0	0	0	1	1	-1	-1
χ₁₀	4	-4	0	0	0	0	0	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₁₁	4	-4	0	0	0	0	0	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$

Here i is the imaginary unit (i² = −1).

Relationships

derived subgroup of	#5932, #5933, #5934, …
maximal subgroup of	B(4,𝔽₂)⋊C₃, #458⋊C₃, #5932, …
maximal normal subgroup of	B(4,𝔽₂)⋊C₃, #458⋊C₃, #5932, …
maximal quotient of	#543⋊C₃, Q₈⋊SL(2,𝔽₃), #5465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 204);