#89894

order	448
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #405

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂×C₄²
Fitting	C₂×C₇×M₁₆	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

130 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)(12,14,16,24,17,22,15,23)(13,18,20,27,21,25,19,26);(14,22)(18,25)(23,24)(26,27);(2,3)(4,5)(6,7)(8,9,10,11)(12,13,15,19,17,21,16,20)(14,18,23,26,22,25,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726451, #11726471, #11726492, …
maximal normal subgroup of	#11726451, #11726471, #11726492, …
maximal quotient of	#11727558, #11727560, #11727564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 656);