C₄⋊Dic₂₅ #87719

order	400
isoclinism	isoclinic class of Dic₅₀
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

C₄⋊Dic₂₅

Splittings

Sylow

C₂₅ ⋊ C₄⋊C₄

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₅
commutator	C₂×C₂₅	C₂×C₄
Frattini, socle, prime radical	C₂²×C₅	C₂×D₅
Fitting	C₂×C₄×C₂₅	C₂
cosocle	C₂²×C₂₅	C₂²
hypercenter	C₂×C₄	D₂₅
hypocenter	C₂₅	C₄⋊C₄
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 2 generators on 33 points:

(1,21,12,24,9,7,14,5,23,4,16,6,22,18,13,8,2,25,10,20,3,15,19,11,17)(30,31,32,33);(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(26,27,28,29)(31,33)

Polycyclic presentation:

2^4,5^2|a^2=c,b^2=d,e^5=f^4,b^a=bd,e^a=e^4f,f^a=f^4

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11716355, #11716356, #11716365, …
maximal normal subgroup of	#11716355, #11716356, #11716365, …
maximal quotient of	C₄⋊#6014, #11716361, #11716366, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 13);

C4⋊Dic25 #87719