#89469

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ #488

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×C₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄×C₈	D₇
hypocenter	C₇	#488
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#89469, #11735908, #11735914, …
maximal normal subgroup of	C₂×#89469, #11735908, #11735914, …
maximal quotient of	#11726384, #11726391, #11726392, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 231);