C₂×D₆₄₁₇ #38787281

order	25668
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₂₈₃₄
direct product	C₂×D₆₄₁₇
unambiguous semidirect product	(C₂×C₉)⋊D₇₁₃
unambiguous semidirect product	(C₂×C₂₃)⋊D₂₇₉
unambiguous semidirect product	(C₂×C₃₁)⋊D₂₀₇

Splittings

Fitting	C₂×C₉×C₂₃×C₃₁ ⋊ C₂
derived, Hall	C₉×C₂₃×C₃₁ ⋊ C₂²
Hall	C₂₃×C₃₁ ⋊ C₂×D₉
Hall	C₉×C₃₁ ⋊ C₂×D₂₃
Hall	C₉×C₂₃ ⋊ C₂×D₃₁
Sylow	C₃₁ ⋊ C₂×D₂₀₇
Sylow	C₂₃ ⋊ C₂×D₂₇₉
Sylow	C₉ ⋊ C₂×D₇₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₄₁₇
commutator, cosocle, hypocenter	C₉×C₂₃×C₃₁	C₂²
Frattini	C₃	C₂×D₂₁₃₉
Fitting	C₂×C₉×C₂₃×C₃₁	C₂
socle	C₂×C₃×C₂₃×C₃₁	D₃
LCS	G ▷ C₉×C₂₃×C₃₁
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 12834

A minimal faithful permutation representation — 2 generators on 65 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);(4,11)(5,10)(6,9)(7,8)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Conjugacy classes & character table

6420 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₅₆₆₈
maximal normal subgroup of	D₂₅₆₆₈
maximal quotient of	Dic₁₂₈₃₄, D₂₅₆₆₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25668, 28);

C2×D6417 #38787281