#89639

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#106 ⋊ C₂
Sylow	C₇ ⋊ #485

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#106	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂²×D₇	D₄
hypercenter	#106	D₇
hypocenter	C₇	#485
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) = 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)(12,15)(16,18,19,25,20,26,27,31)(17,22,23,30,21,24,28,29);(8,10,11,14)(9,12,13,15)(17,21)(18,25)(19,27)(22,29)(24,30)(26,31);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,24)(19,23)(20,21)(22,26)(25,29)(27,28)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#89639, #11736557, #11736570, …
maximal normal subgroup of	C₂×#89639, #11736557, #11736570, …
maximal quotient of	#11726548, #11726567, #11726569, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 401);