C₂×D₄₂₂₁ #30683095

order	16884
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₈₄₄₂
direct product	C₂×D₄₂₂₁
unambiguous semidirect product	(C₂×C₇)⋊D₆₀₃
unambiguous semidirect product	(C₂×C₉)⋊D₄₆₉
unambiguous semidirect product	(C₂×C₆₇)⋊D₆₃

Splittings

Fitting	C₂×C₇×C₉×C₆₇ ⋊ C₂
derived, Hall	C₇×C₉×C₆₇ ⋊ C₂²
Hall	C₉×C₆₇ ⋊ C₂×D₇
Hall	C₇×C₆₇ ⋊ C₂×D₉
Sylow	C₆₇ ⋊ C₂×D₆₃
Hall	C₇×C₉ ⋊ C₂×D₆₇
Sylow	C₉ ⋊ C₂×D₄₆₉
Sylow	C₇ ⋊ C₂×D₆₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₂₂₁
commutator, cosocle, hypocenter	C₇×C₉×C₆₇	C₂²
Frattini	C₃	C₂×D₁₄₀₇
Fitting	C₂×C₇×C₉×C₆₇	C₂
socle	C₂×C₃×C₇×C₆₇	D₃
LCS	G ▷ C₇×C₉×C₆₇
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 8442

A minimal faithful permutation representation — 2 generators on 85 points:

(1,2)(3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(4,11)(5,10)(6,9)(7,8)(13,18)(14,17)(15,16)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

4224 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₆₈₈₄, C₂×D₁₂₆₆₃
maximal normal subgroup of	D₁₆₈₈₄
maximal quotient of	Dic₈₄₄₂, D₁₆₈₈₄, C₂×D₁₂₆₆₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16884, 64);

C2×D4221 #30683095