Dic₉×C₄₉ #13183483

order	1764
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₄₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₉×C₄₉
unambiguous semidirect product	C₉⋊(C₄×C₄₉)

Splittings

derived, Sylow	C₉ ⋊ C₄×C₄₉
Hall	C₉×C₄₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄₉	D₉
commutator, hypocenter	C₉	C₄×C₄₉
Frattini, socle, prime radical	C₂×C₃×C₇	D₃×C₇
Fitting	C₂×C₉×C₄₉	C₂
cosocle	C₂×C₇×C₉	C₂×C₇
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₄₉

27 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 62
transitive degree	μ_t(G) = 1764

A minimal faithful permutation representation — 2 generators on 62 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₁×Dic₉×C₄₉, C₄₉×Dic₉₉, C₁₃×Dic₉×C₄₉, …
maximal normal subgroup of	C₁₁×Dic₉×C₄₉, C₁₃×Dic₉×C₄₉, C₁₇×Dic₉×C₄₉, …
maximal quotient of	C₁₁×Dic₉×C₄₉, C₄₉×Dic₉₉, C₁₃×Dic₉×C₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1764, 3);

Dic9×C49 #13183483