#89666

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#108 ⋊ C₂
Sylow	C₇ ⋊ #444

Subgroups

	subgroup	G/H
center	C₄	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	C₄×D₄
prime radical	#6467	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#108	D₇
hypocenter	C₇	#444
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

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

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

Polycyclic presentation:

2^6,7|b^2=e,c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,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	#11727008, #11727018, #11727063, …
maximal normal subgroup of	#11727008, #11727018, #11727063, …
maximal quotient of	#11726271, #11726272, #11726273, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 428);