#88172

order	416
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₁₃×Q₁₆ ⋊ C₂
Sylow	C₁₃ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₁₃×Q₁₆	C₂
socle	C₂×C₁₃	C₂×D₄
prime radical	#6131	C₂
hypercenter	Q₁₆	D₁₃
hypocenter	C₁₃	#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 29 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#88172, #11719471, #11719473, …
maximal normal subgroup of	C₂×#88172, #11719471, #11719473, …
maximal quotient of	#11718450, #11718452, #11718456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 139);