#89770

order	448
isoclinism	isoclinic class of #1141
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #427

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#427
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4,6,7,5,3)(8,9,10,11)(13,18)(14,21)(15,23)(17,24)(19,20)(25,26);(12,14)(13,17)(15,22)(16,21)(18,26)(19,24)(20,25)(23,27);(2,3)(4,5)(6,7)(12,13,15,19,16,20,23,18)(14,17,22,24,21,25,27,26)

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726453, #11726464, #11726492, …
maximal normal subgroup of	#11726453, #11726464, #11726492, …
maximal quotient of	#11727222, C₄×#6382, C₄×#6381, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 532);