C₂×C₅×C₇⋊(C₁₃⋊D₁₃) #37556909

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

Aliases

direct product	C₂×C₅×C₇⋊(C₁₃⋊D₁₃)
unambiguous semidirect product	(C₂×C₇)⋊(C₅×C₁₃⋊D₁₃)
unambiguous semidirect product	(C₂×C₁₃)⋊(C₅×D₉₁)

Splittings

Fitting, socle	C₂×C₅×C₇×C₁₃² ⋊ C₂
derived, Hall	C₇×C₁₃² ⋊ C₂²×C₅
Hall	C₅×C₇×C₁₃² ⋊ C₂²
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	C₂×C₅	C₇⋊(C₁₃⋊D₁₃)
commutator, cosocle, hypocenter	C₇×C₁₃²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇×C₁₃²	C₂
LCS	G ▷ C₇×C₁₃²
UCS	C₁ ◁ C₂×C₅

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 11830

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(4,5)(6,7)(8,9,10,11,12)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38);(1,2,4,6,7,5,3)(26,27,29,31,33,35,37,38,36,34,32,30,28)(39,40);(13,14,16,18,20,22,24,25,23,21,19,17,15)

Conjugacy classes & character table

5930 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23660, 139);

C2×C5×C7⋊(C13⋊D13) #37556909