C₂²⋊C₉×C₅₉ #14523767

order	2124
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₉×C₅₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²⋊C₉×C₅₉
unambiguous semidirect product	C₂²⋊(C₉×C₅₉)
unambiguous semidirect product	(C₂²×C₅₉)⋊C₉

Splittings

derived, Sylow	C₂² ⋊ C₉×C₅₉
Hall	C₂²×C₅₉ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅₉	A₄
commutator, hypocenter	C₂²	C₉×C₅₉
Frattini	C₃	A₄×C₅₉
Fitting, socle, prime radical	C₂²×C₃×C₅₉	C₃
cosocle	C₂²×C₃	C₃×C₅₉
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₅₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 1062

A minimal faithful permutation representation — 2 generators on 72 points:

(2,3,4)(5,6,8,7,9,11,10,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72);(1,4)(2,3)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×C₂²⋊C₉×C₅₉, C₅₉×C₂²⋊(C₇⋊C₉), C₁₁×C₂²⋊C₉×C₅₉, …
maximal normal subgroup of	C₇×C₂²⋊C₉×C₅₉, C₁₁×C₂²⋊C₉×C₅₉, C₁₃×C₂²⋊C₉×C₅₉, …
maximal quotient of	C₇×C₂²⋊C₉×C₅₉, C₅₉×C₂²⋊(C₇⋊C₉), C₁₁×C₂²⋊C₉×C₅₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2124, 7);

C22⋊C9×C59 #14523767