#88059

order	416
isoclinism	isoclinic class of #6095
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃ ⋊ #108

Subgroups

	subgroup	G/H
center	C₄	D₅₂
commutator	C₄×C₁₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₃
Fitting	C₂×C₈×C₁₃	C₂
socle	C₂×C₁₃	C₄⋊C₄
prime radical	C₂²×C₁₃	Q₈
cosocle	C₂×C₄×C₁₃	C₂²
hypercenter	C₂×C₈	D₁₃
hypocenter	C₁₃	#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

30 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)(14,16,18,25,19,26,17,24)(15,20,21,23,22,28,27,29);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,17,21,19,22,18,27)(16,23,24,28,26,29,25,20)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718079, #11718085, #11718098, …
maximal normal subgroup of	#11718079, #11718085, #11718098, …
maximal quotient of	#11718049, #11718050, #11718144, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 26);