#89960

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 ⋊ #467

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	#467
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

114 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)(13,17,19,25)(15,23,24,21)(18,26,27,20);(12,13,15,18,16,19,24,27)(14,20,21,25,22,26,23,17);(2,3)(4,5)(6,7)(8,9)(10,11)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726968, #11726981, #11726982, …
maximal normal subgroup of	#11726968, #11726981, #11726982, …
maximal quotient of	#11727998, #11728002, #11728008, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 722);