C₇×C₁₃×Dic₁₁ #16860702

order	4004
isoclinism	isoclinic class of D₁₁
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₁₃×Dic₁₁
unambiguous semidirect product	C₁₁⋊(C₄×C₇×C₁₃)

Splittings

derived, Sylow	C₁₁ ⋊ C₄×C₇×C₁₃
Hall	C₇×C₁₁×C₁₃ ⋊ C₄
Hall	C₁₁×C₁₃ ⋊ C₄×C₇
Hall	C₇×C₁₁ ⋊ C₄×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇×C₁₃	D₁₁
commutator, hypocenter	C₁₁	C₄×C₇×C₁₃
Frattini	C₂	C₇×C₁₃×D₁₁
Fitting, socle, prime radical	C₂×C₇×C₁₁×C₁₃	C₂
cosocle	C₂×C₁₁	C₂×C₇×C₁₃
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂×C₇×C₁₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 4004

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

1274 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₇×C₁₃×Dic₁₁, C₇×C₁₃×Dic₃₃, C₇×C₁₃⋊C₃×Dic₁₁, …
maximal normal subgroup of	C₃×C₇×C₁₃×Dic₁₁, C₇×C₁₃⋊C₃×Dic₁₁, C₁₃×ΓL(1,𝔽₈)×Dic₁₁, …
maximal quotient of	C₃×C₇×C₁₃×Dic₁₁, C₇×C₁₃×Dic₃₃, C₅×C₇×C₁₃×Dic₁₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4004, 3);

C7×C13×Dic11 #16860702