#89901

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×M₁₆ ⋊ C₂
Sylow	C₇ ⋊ #408

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₂×C₇×M₁₆	C₂
prime radical	C₂×#1157	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#408
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

158 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)(9,11)(12,14,15,21,16,20,22,26)(13,18,19,25,17,23,24,27);(13,17)(14,20)(19,24)(21,26);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,18)(15,19)(16,17)(20,23)(21,25)(22,24)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725337, #11726469, #11726481, …
maximal normal subgroup of	#11725337, #11726469, #11726481, …
maximal quotient of	#11727584, #11727591, #11727592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 663);