#88104

order	416
isoclinism	isoclinic class of #88101
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	C₂	C₄⋊#261
commutator	C₄×C₁₃	C₂×C₄
Frattini	C₄	C₂×#261
Fitting	C₈×C₁₃	C₄
socle	C₂×C₁₃	C₄⋊C₄
prime radical	C₂×D₁₃	Q₈
cosocle	C₄×D₁₃	C₂²
hypercenter	C₈	#261
hypocenter	C₁₃	#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ 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:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718223, #11718224, #11718277, …
maximal normal subgroup of	#11718223, #11718224, #11718277, …
maximal quotient of	#11718254, #11718255, #11718268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 71);