#89954

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #462

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#462
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2^6,7|a^2=e,b^2=f,c^2=f,d^2=f,c^a=ce,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	#11737144, #11737145, #11737147, …
maximal normal subgroup of	#11737144, #11737145, #11737147, …
maximal quotient of	#11727975, #11727983, #11727987, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 716);