C₂×D₂₇₄₅ #24926720

order	10980
automorphism	Aut(G) order 7905600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₅₄₉₀
direct product	C₂×D₂₇₄₅
unambiguous semidirect product	(C₂×C₅)⋊D₅₄₉
unambiguous semidirect product	(C₂×C₉)⋊D₃₀₅
unambiguous semidirect product	(C₂×C₆₁)⋊D₄₅

Splittings

Fitting	C₂×C₅×C₉×C₆₁ ⋊ C₂
derived, Hall	C₅×C₉×C₆₁ ⋊ C₂²
Hall	C₉×C₆₁ ⋊ C₂×D₅
Hall	C₅×C₆₁ ⋊ C₂×D₉
Sylow	C₆₁ ⋊ C₂×D₄₅
Hall	C₅×C₉ ⋊ C₂×D₆₁
Sylow	C₉ ⋊ C₂×D₃₀₅
Sylow	C₅ ⋊ C₂×D₅₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₇₄₅
commutator, cosocle, hypocenter	C₅×C₉×C₆₁	C₂²
Frattini	C₃	C₂×D₉₁₅
Fitting	C₂×C₅×C₉×C₆₁	C₂
socle	C₂×C₃×C₅×C₆₁	D₃
LCS	G ▷ C₅×C₉×C₆₁
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 5490

A minimal faithful permutation representation — 2 generators on 77 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);(4,11)(5,10)(6,9)(7,8)(13,16)(14,15)(18,77)(19,76)(20,75)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Conjugacy classes & character table

2748 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₀₉₈₀, C₂×D₈₂₃₅, C₂×D₁₃₇₂₅
maximal normal subgroup of	D₁₀₉₈₀
maximal quotient of	D₁₀₉₈₀, Dic₅₄₉₀, C₂×D₈₂₃₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10980, 70);

C2×D2745 #24926720