#88205

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

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting, prime radical	C₁₃×#41	C₂
socle	C₂×C₁₃	C₂×D₄
hypercenter	#41	D₁₃
hypocenter	C₁₃	#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

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

Polycyclic presentation:

2^5,13|a^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	#11718486, #11718487, #11718802, …
maximal normal subgroup of	#11718486, #11718487, #11718802, …
maximal quotient of	#11718656, #11718657, #11718665, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 172);