#88169

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

Aliases

No aliases for this group.

Splittings

Fitting	C₁₃×SD₁₆ ⋊ C₂
Sylow	C₁₃ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₁₃×SD₁₆	C₂
socle	C₂×C₁₃	C₂×D₄
prime radical	#6129	C₂
hypercenter	SD₁₆	D₁₃
hypocenter	C₁₃	#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

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

Polycyclic presentation:

2^5,13|b^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×#88169, #11719467, #11719473, …
maximal normal subgroup of	C₂×#88169, #11719467, #11719473, …
maximal quotient of	#11718418, #11718420, #11718426, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 136);