#89687

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×SD₃₂ ⋊ C₂
Sylow	C₇ ⋊ #510

Subgroups

	subgroup	G/H
center	C₂	D₇×D₈
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₇×SD₃₂	C₂
socle	C₂×C₇	C₂×D₈
prime radical	#6475	C₂
hypercenter	SD₃₂	D₇
hypocenter	C₇	#510
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,4,6,7,5,3)(9,15)(10,18)(11,22)(12,23)(14,26)(16,33)(19,20)(21,37)(24,38)(25,36)(27,35)(28,39)(29,32)(30,34);(8,10,13,21)(9,14,15,29)(11,18,24,37)(12,25,23,20)(16,32,30,26)(17,35,31,28)(19,38,36,22)(27,33,39,34);(2,3)(4,5)(6,7)(8,9)(10,14)(11,16)(12,17)(13,15)(18,32)(19,27)(20,28)(21,29)(22,34)(23,31)(24,30)(25,35)(26,37)(33,38)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#89687, #11739469, #11739475, …
maximal normal subgroup of	C₂×#89687, #11739469, #11739475, …
maximal quotient of	#11726930, #11726932, #11726939, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 449);