C₂₃⋊C₁₆ #67259

order	368
isoclinism	isoclinic class of D₂₃
automorphism	?
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₂₃⋊C₁₆

Splittings

derived, Sylow

C₂₃ ⋊ C₁₆

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₂₃
commutator, hypocenter	C₂₃	C₁₆
Fitting, cosocle	C₈×C₂₃	C₂
socle, prime radical	C₂×C₂₃	C₈
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₈

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 368

A minimal faithful permutation representation — 2 generators on 39 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25,26,29,27,30,32,35,28,31,33,36,34,37,38,39);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)

Polycyclic presentation:

2^4,23|a^2=b,b^2=c,c^2=d,e^a=e^22

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×D₂₃, #10624980, C₂×C₂₃⋊C₁₆, …
maximal normal subgroup of	C₁₆×D₂₃, #10624980, C₂×C₂₃⋊C₁₆, …
maximal quotient of	C₂₃⋊C₃₂, C₂×C₂₃⋊C₁₆, C₃×C₂₃⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 1);

C23⋊C16 #67259