#23778454⋊C₃ #44477885

order	30420
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	#23778454⋊C₃
unambiguous semidirect product	#23778456⋊C₃
unambiguous semidirect product	#23778464⋊C₃

Splittings

Fitting, socle, derived, Hall	C₅×C₁₃² ⋊ C₃²×C₄
Hall	C₅×C₁₃⋊C₃² ⋊ C₄
Hall	#16810003 ⋊ C₃²
Hall	C₁₃⋊C₃² ⋊ AGL(1,𝔽₅)
Sylow	C₁₃² ⋊ C₃²×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #14523144⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₁₃²	C₃²×C₄
prime radical	C₅⋊((C₁₃⋊C₃)⋊(D₁₃⋊C₃))	C₂
cosocle	C₅⋊(C₁₃⋊D₁₃)	C₂×C₃²
LCS	G ▷ C₅×C₁₃²
UCS	C₁

192 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 195

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

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

Polycyclic presentation:

2,3^2,2,5,13^2|a^2=d,e^a=e^2,f^a=f^8,g^a=g^8,g^b=g^3,f^c=f^3,e^d=e^4,f^d=f^12,g^d=g^12

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 474);