#88196

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×Q₈×C₁₃ ⋊ C₂
Sylow	C₁₃ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₂×Q₈×C₁₃	C₂
socle	C₂×C₁₃	C₂×D₄
prime radical	#6127	C₂
hypercenter	C₂×Q₈	D₁₃
hypocenter	C₁₃	#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718415, #11718416, #11718722, …
maximal normal subgroup of	#11718415, #11718416, #11718722, …
maximal quotient of	#11718625, #11718627, #11718628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 163);