C₂×D₂₇₀₉ #24925487

order	10836
automorphism	Aut(G) order 8192016
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₅₄₁₈
direct product	C₂×D₂₇₀₉
unambiguous semidirect product	(C₂×C₇)⋊D₃₈₇
unambiguous semidirect product	(C₂×C₉)⋊D₃₀₁
unambiguous semidirect product	(C₂×C₄₃)⋊D₆₃

Splittings

Fitting	C₂×C₇×C₉×C₄₃ ⋊ C₂
derived, Hall	C₇×C₉×C₄₃ ⋊ C₂²
Hall	C₉×C₄₃ ⋊ C₂×D₇
Hall	C₇×C₄₃ ⋊ C₂×D₉
Hall	C₇×C₉ ⋊ C₂×D₄₃
Sylow	C₄₃ ⋊ C₂×D₆₃
Sylow	C₉ ⋊ C₂×D₃₀₁
Sylow	C₇ ⋊ C₂×D₃₈₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₇₀₉
commutator, cosocle, hypocenter	C₇×C₉×C₄₃	C₂²
Frattini	C₃	C₂×D₉₀₃
Fitting	C₂×C₇×C₉×C₄₃	C₂
socle	C₂×C₃×C₇×C₄₃	D₃
LCS	G ▷ C₇×C₉×C₄₃
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 5418

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

Conjugacy classes & character table

2712 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₀₈₃₆, C₂×D₈₁₂₇, C₂×D₁₃₅₄₅
maximal normal subgroup of	D₁₀₈₃₆
maximal quotient of	D₁₀₈₃₆, Dic₅₄₁₈, C₂×D₈₁₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10836, 79);

C2×D2709 #24925487