Finite Group Databasebeta

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Cayley diagram

D4×C25 #6023

order200
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C4×C5×D4 (order 160) · Inn(G) = C22 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C22×C25 · Schur multiplier (H2) = C2

Aliases

direct productD4×C25
unambiguous semidirect productC4⋊(C2×C25)
unambiguous semidirect productC22⋊(C2×C25)

Subgroups

subgroupG/H
centerC2×C25C22
commutatorC2C22×C25
Frattini, socle, cosocleC2×C5C22×C5
prime radicalC5×D4C5
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C25 ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(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);(2,4)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#87721, #87722, D4×D25, …
maximal subgroup of#87721, #87722, D8×C25, …
maximal normal subgroup of#87721, #87722, D8×C25, …
maximal quotient ofC22⋊C4×C25, C4⋊C4×C25, D8×C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 10);