C₂×D₅₂₄₇ #35230563

order	20988
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₄₉₄
direct product	C₂×D₅₂₄₇
unambiguous semidirect product	(C₂×C₉)⋊D₅₈₃
unambiguous semidirect product	(C₂×C₁₁)⋊D₄₇₇
unambiguous semidirect product	(C₂×C₅₃)⋊D₉₉

Splittings

Fitting	C₂×C₉×C₁₁×C₅₃ ⋊ C₂
derived, Hall	C₉×C₁₁×C₅₃ ⋊ C₂²
Hall	C₁₁×C₅₃ ⋊ C₂×D₉
Hall	C₉×C₅₃ ⋊ C₂×D₁₁
Hall	C₉×C₁₁ ⋊ C₂×D₅₃
Sylow	C₅₃ ⋊ C₂×D₉₉
Sylow	C₁₁ ⋊ C₂×D₄₇₇
Sylow	C₉ ⋊ C₂×D₅₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₂₄₇
commutator, cosocle, hypocenter	C₉×C₁₁×C₅₃	C₂²
Frattini	C₃	C₂×D₁₇₄₉
Fitting	C₂×C₉×C₁₁×C₅₃	C₂
socle	C₂×C₃×C₁₁×C₅₃	D₃
LCS	G ▷ C₉×C₁₁×C₅₃
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 10494

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

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

Conjugacy classes & character table

5250 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₀₉₈₈, C₂×D₁₅₇₄₁
maximal normal subgroup of	D₂₀₉₈₈
maximal quotient of	Dic₁₀₄₉₄, D₂₀₉₈₈, C₂×D₁₅₇₄₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20988, 22);

C2×D5247 #35230563