#89676

order	448
isoclinism	isoclinic class of D₁₁₂
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×C₁₆ ⋊ C₂
Sylow	C₇ ⋊ #508

Subgroups

	subgroup	G/H
center	C₄	D₅₆
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle	C₂×C₇	C₂×D₈
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	#508
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,4,6,7,5,3)(8,11,22,36,13,24,27,37,14,25,20,33,10,19,12,23)(9,16,26,32,15,29,31,35,18,30,39,38,28,34,17,21);(8,10,14,13)(9,15,18,28)(11,19,25,24)(12,20,27,22)(16,29,30,34)(17,26,31,39)(21,32,35,38)(23,33,37,36);(2,3)(4,5)(6,7)(8,9)(10,15)(11,21)(12,26)(13,28)(14,18)(16,23)(17,22)(19,32)(20,31)(24,38)(25,35)(27,39)(29,33)(30,37)(34,36)

Polycyclic presentation:

2^6,7|b^2=f,c^2=de,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^6

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725386, #11725391, #11725404, …
maximal normal subgroup of	#11725386, #11725391, #11725404, …
maximal quotient of	#11726799, C₄×D₁₁₂, #11726813, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 438);