#89776

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #419

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#419
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 224

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726461, #11726463, #11726466, …
maximal normal subgroup of	#11726461, #11726463, #11726466, …
maximal quotient of	#11727187, #11727199, #11727229, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 538);