C₅×C₇×C₁₁²⋊C₃ #27163203

order	12705
isoclinism	isoclinic class of C₁₁²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₅×C₇ · Schur multiplier (H₂) = C₁₁

Aliases

direct product	C₅×C₇×C₁₁²⋊C₃
unambiguous semidirect product	(C₅×C₁₁²)⋊(C₃×C₇)

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₁² ⋊ C₃
derived, Sylow	C₁₁² ⋊ C₃×C₅×C₇
Hall	C₇×C₁₁² ⋊ C₃×C₅
Hall	C₅×C₁₁² ⋊ C₃×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₇	C₁₁²⋊C₃
commutator, cosocle, hypocenter	C₁₁²	C₃×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₁²	C₃
LCS	G ▷ C₁₁²
UCS	C₁ ◁ C₅×C₇

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1155

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,4)(3,7,15)(5,11,19)(6,13,25)(8,18,10)(9,20,14)(12,24,33)(16,21,29)(17,22,31)(23,32,26)(27,30,28)(34,35,36,37,38,39,40)(41,42,43,44,45);(1,3,8,19,29,32,33,20,30,25,31)(2,14,5,17,12,18,6,26,7,28,16)(4,11,24,13,15,21,9,22,10,23,27)

Conjugacy classes & character table

1505 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	C₅⋊(C₇⋊#10624391)
maximal subgroup of	C₅×C₇×#10624391, C₅×C₇×(C₁₁⋊D₁₁)⋊C₃, C₅×D₇×C₁₁²⋊C₃, …
maximal normal subgroup of	C₅×C₇×#10624391, C₅×C₇×(C₁₁⋊D₁₁)⋊C₃, C₅×D₇×C₁₁²⋊C₃, …
maximal quotient of	C₂×C₅×C₇×C₁₁²⋊C₃, C₅×C₇×C₁₁²⋊C₉, C₃×C₅×C₇×C₁₁²⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12705, 18);

C5×C7×C112⋊C3 #27163203