#89853

order	448
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #487

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#487
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,4,6,7,5,3)(9,13)(10,14)(16,19,20,17)(18,22,21,23);(8,10,11,14)(9,12,13,15)(16,18,20,21)(17,22,19,23);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,21,23,20,19,18,22)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726198, #11726201, #11726206, …
maximal normal subgroup of	#11726198, #11726201, #11726206, …
maximal quotient of	#11727326, #11727328, #11727336, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 615);