#90009

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #413

Subgroups

	subgroup	G/H
center	C₄	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₂×C₇×#41	C₂
socle	C₂×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×#41	D₇
hypocenter	C₇	#413
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725498, #11725499, #11725503, …
maximal normal subgroup of	#11725498, #11725499, #11725503, …
maximal quotient of	#11729325, #11729326, #11729341, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 771);