#88076

order	416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃ ⋊ #101

Subgroups

	subgroup	G/H
center	C₂	C₂²⋊Dic₁₃
commutator, prime radical	C₂²×C₁₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₃
Fitting	C₂×Q₈×C₁₃	C₂
socle	C₂×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄×C₁₃	C₂²
hypercenter	C₂×Q₈	D₁₃
hypocenter	C₁₃	#101
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

38 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)(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,17,20,18,21,26,29)(16,22,23,28,24,19,25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718069, #11718070, #11718134, …
maximal normal subgroup of	#11718069, #11718070, #11718134, …
maximal quotient of	#11718120, #11718122, #11718135, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 43);