Finite Group Databasebeta

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D₁₅₆ #64623

order	312
isoclinism	isoclinic class of Dic₇₈
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₅₆
unambiguous semidirect product	C₄⋊D₃₉

Splittings

Fitting	C₃×C₄×C₁₃ ⋊ C₂
Hall	C₃×C₁₃ ⋊ D₄
Sylow	C₁₃ ⋊ D₁₂
Sylow	C₃ ⋊ D₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₉
commutator, socle, cosocle	C₂×C₃×C₁₃	C₂²
Fitting	C₃×C₄×C₁₃	C₂
hypercenter	C₄	D₃₉
hypocenter	C₃×C₁₃	D₄
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20);(2,4)(6,7)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)

Polycyclic presentation:

2^3,3,13|b^2=c,b^a=bc,d^a=d^2,e^a=e^12

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597596, #10597597, #10597601, …
maximal normal subgroup of	#10597596, #10597597, #10597601, …
maximal quotient of	#10597658, D₃₁₂, Dic₁₅₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 39);