#89517

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#100 ⋊ C₂
Sylow	C₇ ⋊ #413

Subgroups

	subgroup	G/H
center	C₂	D₇×C₂²⋊C₄
commutator	C₂²×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#100	C₂
socle	C₂×C₇	C₂×C₂²⋊C₄
prime radical	C₂×#1159	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#100	D₇
hypocenter	C₇	#413
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^6,7|c^2=d,d^2=f,b^a=bf,g^a=g^6,c^b=ce,d^b=df,e^c=ef

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726890, #11726891, #11726898, …
maximal normal subgroup of	#11726890, #11726891, #11726898, …
maximal quotient of	#11725854, #11725856, #11725859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 279);