C₄⋊Dic₃₁ #92477

order	496
isoclinism	isoclinic class of Dic₆₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

C₄⋊Dic₃₁

Splittings

Sylow

C₃₁ ⋊ C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃₁
commutator	C₂×C₃₁	C₂×C₄
Fitting	C₂×C₄×C₃₁	C₂
socle, prime radical, cosocle	C₂²×C₃₁	C₂²
hypercenter	C₂×C₄	D₃₁
hypocenter	C₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 496

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(36,37,38,39);(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)(37,39)

Polycyclic presentation:

2^4,31|a^2=c,b^2=d,b^a=bd,e^a=e^30

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11758395, #11758396, #11758404, …
maximal normal subgroup of	#11758395, #11758396, #11758404, …
maximal quotient of	C₄⋊(C₃₁⋊C₈), #11758405, #11758406, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(496, 12);

C4⋊Dic31 #92477