#89626

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #475

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#106	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#106	D₇
hypocenter	C₇	#475
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11736569, #11736571, #11736575, …
maximal normal subgroup of	#11736569, #11736571, #11736575, …
maximal quotient of	#11726542, #11726554, #11726564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 388);