#89461

order	448
isoclinism	isoclinic class of Dic₁₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ #405

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₇
commutator	C₂×C₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×C₈	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂²×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×C₈	D₇
hypocenter	C₇	#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726805, #11726807, #11726826, …
maximal normal subgroup of	#11726805, #11726807, #11726826, …
maximal quotient of	C₈×#1131, #11725801, #11725802, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 223);