#89951

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 ⋊ #510

Subgroups

	subgroup	G/H
center	C₂	C₂×#1141
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle	C₂×C₇	C₂×D₈
prime radical	#6467	C₂
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#510
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11727018, #11727026, #11727027, …
maximal normal subgroup of	#11727018, #11727026, #11727027, …
maximal quotient of	#11728350, #11728352, #11728353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 713);