#89861

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×C₄⋊Q₈ ⋊ C₂
Sylow	C₇ ⋊ #489

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊Q₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#489
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11728998, #11729000, #11729005, …
maximal normal subgroup of	#11728998, #11729000, #11729005, …
maximal quotient of	#11727329, #11727335, #11727339, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 623);