C₁₃×D₄₇₇ #27137145

order	12402
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×D₄₇₇
unambiguous semidirect product	C₉⋊(C₁₃×D₅₃)

Splittings

Fitting, Hall	C₉×C₁₃×C₅₃ ⋊ C₂
derived, Hall	C₉×C₅₃ ⋊ C₂×C₁₃
Hall	C₁₃×C₅₃ ⋊ D₉
Hall	C₉×C₁₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ C₁₃×D₉
Sylow	C₉ ⋊ C₁₃×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	D₄₇₇
commutator, cosocle, hypocenter	C₉×C₅₃	C₂×C₁₃
Frattini	C₃	C₁₃×D₁₅₉
Fitting	C₉×C₁₃×C₅₃	C₂
socle	C₃×C₁₃×C₅₃	D₃
LCS	G ▷ C₉×C₅₃
UCS	C₁ ◁ C₁₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6201

A minimal faithful permutation representation — 2 generators on 75 points:

(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)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(55,57)(56,60)(58,61)(59,62)(63,64,65,66,67,68,69,70,71,72,73,74,75);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,59,57,56,61,58,60,55,62)

Conjugacy classes & character table

3120 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃×D₉×D₅₃, D₁₃×D₄₇₇, D₁₁₇⋊D₅₃, …
maximal normal subgroup of	C₁₃×D₉×D₅₃, D₁₃×D₄₇₇, D₁₁₇⋊D₅₃, …
maximal quotient of	C₁₃×Dic₄₇₇, C₂×C₁₃×D₄₇₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 8);

C13×D477 #27137145