#89828

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

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #488

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#124	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×C₇×D₄	C₂²
hypercenter	#124	D₇
hypocenter	C₇	#488
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,4,6,7,5,3)(9,13)(10,14)(17,22)(18,25)(19,27)(21,28)(23,24)(29,30);(8,10,11,14)(9,12,13,15)(16,18)(17,21)(19,26)(20,25)(22,30)(23,28)(24,29)(27,31);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,23,20,24,27,22)(18,21,26,28,25,29,31,30)

Polycyclic presentation:

2^6,7|a^2=e,b^2=d,e^2=f,c^a=cde,g^a=g^6,c^b=cdf,e^c=ef

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725349, #11725353, #11737961, …
maximal normal subgroup of	#11725349, #11725353, #11737961, …
maximal quotient of	#11727296, #11727301, #11727308, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 590);