C₂×D₇₈₃₉ #44535634

order	31356
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₅₆₇₈
direct product	C₂×D₇₈₃₉
unambiguous semidirect product	(C₂×C₉)⋊D₈₇₁
unambiguous semidirect product	(C₂×C₁₃)⋊D₆₀₃
unambiguous semidirect product	(C₂×C₆₇)⋊D₁₁₇

Splittings

Fitting	C₂×C₉×C₁₃×C₆₇ ⋊ C₂
derived, Hall	C₉×C₁₃×C₆₇ ⋊ C₂²
Hall	C₁₃×C₆₇ ⋊ C₂×D₉
Hall	C₉×C₆₇ ⋊ C₂×D₁₃
Hall	C₉×C₁₃ ⋊ C₂×D₆₇
Sylow	C₆₇ ⋊ C₂×D₁₁₇
Sylow	C₁₃ ⋊ C₂×D₆₀₃
Sylow	C₉ ⋊ C₂×D₈₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₈₃₉
commutator, cosocle, hypocenter	C₉×C₁₃×C₆₇	C₂²
Frattini	C₃	C₂×D₂₆₁₃
Fitting	C₂×C₉×C₁₃×C₆₇	C₂
socle	C₂×C₃×C₁₃×C₆₇	D₃
LCS	G ▷ C₉×C₁₃×C₆₇
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 15678

A minimal faithful permutation representation — 2 generators on 91 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(4,11)(5,10)(6,9)(7,8)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Conjugacy classes & character table

7842 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁₃₅₆
maximal normal subgroup of	D₃₁₃₅₆
maximal quotient of	Dic₁₅₆₇₈, D₃₁₃₅₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31356, 70);

C2×D7839 #44535634