D₁₃⋊C₁₆ #88097

order	416
isoclinism	isoclinic class of #261
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

D₁₃⋊C₁₆

Splittings

derived, Sylow

C₁₃ ⋊ C₂×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₈	#261
commutator, hypocenter	C₁₃	C₂×C₁₆
Frattini	C₄	C₂×#261
Fitting	C₈×C₁₃	C₄
socle	C₂×C₁₃	C₂×C₈
prime radical	C₂×D₁₃	C₈
cosocle	#1055	C₂²
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₈

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

Polycyclic presentation:

2^5,13|a^2=c,b^2=d,c^2=d,d^2=e,f^a=f^8,f^c=f^12

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×#261, #11718216, #11718276, …
maximal normal subgroup of	C₁₆×#261, #11718216, #11718276, …
maximal quotient of	D₁₃⋊C₃₂, #11718214, #11718257, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 64);

D13⋊C16 #88097