C₂×D₅₆₃₅ #36376758

order	22540
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₁₂₇₀
direct product	C₂×D₅₆₃₅
unambiguous semidirect product	(C₂×C₅)⋊D₁₁₂₇
unambiguous semidirect product	(C₂×C₂₃)⋊D₂₄₅
unambiguous semidirect product	(C₂×C₄₉)⋊D₁₁₅

Splittings

Fitting	C₂×C₅×C₂₃×C₄₉ ⋊ C₂
derived, Hall	C₅×C₂₃×C₄₉ ⋊ C₂²
Hall	C₂₃×C₄₉ ⋊ C₂×D₅
Hall	C₅×C₄₉ ⋊ C₂×D₂₃
Hall	C₅×C₂₃ ⋊ C₂×D₄₉
Sylow	C₄₉ ⋊ C₂×D₁₁₅
Sylow	C₂₃ ⋊ C₂×D₂₄₅
Sylow	C₅ ⋊ C₂×D₁₁₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₆₃₅
commutator, cosocle, hypocenter	C₅×C₂₃×C₄₉	C₂²
Frattini	C₇	C₂×D₈₀₅
Fitting	C₂×C₅×C₂₃×C₄₉	C₂
socle	C₂×C₅×C₇×C₂₃	D₇
LCS	G ▷ C₅×C₂₃×C₄₉
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 11270

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

Conjugacy classes & character table

5638 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₅₄₀
maximal normal subgroup of	D₂₂₅₄₀
maximal quotient of	Dic₁₁₂₇₀, D₂₂₅₄₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22540, 21);

C2×D5635 #36376758