#88068

order	416
isoclinism	isoclinic class of #88066
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₁₃×Q₁₆ ⋊ C₂
Sylow	C₁₃ ⋊ SD₃₂

Subgroups

	subgroup	G/H
center	C₂	#6104
commutator, cosocle	C₈×C₁₃	C₂²
Frattini	C₈	C₂×D₁₃
Fitting	C₁₃×Q₁₆	C₂
socle	C₂×C₁₃	D₈
prime radical	D₁₀₄	C₂
hypercenter	Q₁₆	D₁₃
hypocenter	C₁₃	SD₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,15,18,21)(16,23,25,29)(17,26,24,20)(19,27,28,22);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(15,19)(16,22)(17,24)(20,29)(21,28)(23,26)(25,27)

Polycyclic presentation:

2^5,13|b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^12,c^b=cd,d^b=de

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₃×SD₃₂, #11718575, #11718579, …
maximal normal subgroup of	D₁₃×SD₃₂, #11718575, #11718579, …
maximal quotient of	#11718082, #11718083, #11718156, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 35);