#89961

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×Q₁₆ ⋊ C₂
Sylow	C₇ ⋊ #470

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂²×D₇	D₄
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#470
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

110 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)(9,11)(12,14,16,22)(13,18,17,25)(15,23,24,21)(19,27,26,20);(12,13,15,19,16,17,24,26)(14,20,21,25,22,27,23,18);(2,3)(4,5)(6,7)(8,9)(10,11)(13,17)(18,25)(19,26)(20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11727021, #11729276, #11737171, …
maximal normal subgroup of	#11727021, #11729276, #11737171, …
maximal quotient of	#11727997, #11728001, #11728010, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 723);