#89690

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×Q₃₂ ⋊ C₂
Sylow	C₇ ⋊ #510

Subgroups

	subgroup	G/H
center	C₂	D₇×D₈
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₇×Q₃₂	C₂
socle	C₂×C₇	C₂×D₈
prime radical	#6482	C₂
hypercenter	Q₃₂	D₇
hypocenter	C₇	#510
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

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

Polycyclic presentation:

2^6,7|b^2=f,c^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×#89690, #11739473, #11739475, …
maximal normal subgroup of	C₂×#89690, #11739473, #11739475, …
maximal quotient of	#11726962, #11726963, #11726964, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 452);