#88113

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⋊#261
commutator, prime radical	C₂²×C₁₃	C₂×C₄
Frattini	C₂²	C₂×#261
Fitting	C₂×C₄×C₁₃	C₄
socle	C₂×C₁₃	C₂²⋊C₄
cosocle	C₂×Dic₁₃	C₂²
hypercenter	C₂×C₄	#261
hypocenter	C₁₃	#101
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

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:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718227, #11718228, #11718295, …
maximal normal subgroup of	#11718227, #11718228, #11718295, …
maximal quotient of	#11718242, #11718245, #11718285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 80);