Dic₂₅ #1033

order	100
isoclinism	isoclinic class of D₂₅
automorphism	Aut(G) = C₂×AGL(1,ℤ/25) (order 1000) · Inn(G) = D₂₅ · Out(G) = C₂²×C₅
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₅

Splittings

derived, Sylow

C₂₅ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₅
commutator, hypocenter	C₂₅	C₄
Frattini, socle, prime radical	C₂×C₅	D₅
Fitting, cosocle	C₂×C₂₅	C₂
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₂

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 29 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);(1,3,2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)

Polycyclic presentation:

2^2,5^2|a^2=b,c^5=d^4,c^a=c^4d,d^a=d^4

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6016, Dic₅₀, C₄×D₂₅, …
maximal normal subgroup of	#6016, Dic₅₀, C₄×D₂₅, …
maximal quotient of	#6014, C₂×Dic₂₅, C₃×Dic₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(100, 1);

Dic25 #1033