C₅×D₇×C₁₃⋊D₁₃ #37556856

order	23660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₁₃

Aliases

direct product	C₅×D₇×C₁₃⋊D₁₃
unambiguous semidirect product	(C₁₃×D₇)⋊(C₅×D₁₃)

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₃² ⋊ C₂²
derived, Hall	C₇×C₁₃² ⋊ C₂²×C₅
Hall	C₅×C₁₃² ⋊ C₂×D₇
Sylow	C₁₃² ⋊ C₂×C₅×D₇
Hall	C₅×C₇ ⋊ C₂×C₁₃⋊D₁₃
Sylow	C₇ ⋊ C₂×C₅×C₁₃⋊D₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₇×C₁₃⋊D₁₃
commutator, cosocle, hypocenter	C₇×C₁₃²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₃²	C₂²
LCS	G ▷ C₇×C₁₃²
UCS	C₁ ◁ C₅

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 5915

A minimal faithful permutation representation — 3 generators on 38 points:

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

Conjugacy classes & character table

2150 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23660, 86);

C5×D7×C13⋊D13 #37556856