C₂₃⋊C₈ #4400

order	184
isoclinism	isoclinic class of D₂₃
automorphism	Aut(G) = C₂²×AGL(1,𝔽₂₃) (order 2024) · Inn(G) = D₂₃ · Out(G) = C₂²×C₁₁
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₂₃⋊C₈

Splittings

derived, Sylow

C₂₃ ⋊ C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₂₃
commutator, hypocenter	C₂₃	C₈
Fitting, cosocle	C₄×C₂₃	C₂
socle, prime radical	C₂×C₂₃	C₄
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₄

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 184

A minimal faithful permutation representation — 2 generators on 31 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25,26,28,27,29,30,31);(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:

2^3,23|a^2=b,b^2=c,d^a=d^22

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₂₃, #67262, C₂×C₂₃⋊C₈, …
maximal normal subgroup of	C₈×D₂₃, #67262, C₂×C₂₃⋊C₈, …
maximal quotient of	C₂₃⋊C₁₆, C₂×C₂₃⋊C₈, C₃×C₂₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(184, 1);

C23⋊C8 #4400