C₁₁×C₅⋊(C₅⋊(C₇⋊D₇)) #41004012

order	26950
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₅×C₇

Aliases

direct product	C₁₁×C₅⋊(C₅⋊(C₇⋊D₇))
unambiguous semidirect product	C₅⋊(C₁₁×C₅⋊(C₇⋊D₇))
unambiguous semidirect product	C₇⋊(C₁₁×C₅⋊D₃₅)

Splittings

Fitting, socle, Hall	C₅²×C₇²×C₁₁ ⋊ C₂
derived, Hall	C₅²×C₇² ⋊ C₂×C₁₁
Hall	C₇²×C₁₁ ⋊ C₅⋊D₅
Hall	C₅²×C₁₁ ⋊ C₇⋊D₇
Sylow	C₇² ⋊ C₁₁×C₅⋊D₅
Sylow	C₅² ⋊ C₁₁×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	C₅⋊(C₅⋊(C₇⋊D₇))
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₅²×C₇²×C₁₁	C₂
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₁₁

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 13475

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

6754 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26950, 79);

C11×C5⋊(C5⋊(C7⋊D7)) #41004012