Finite Group Databasebeta

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C22×C4×C5×C125 #72188677

order10000
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) order 1920000
homologyAbelianization (H1) = G · Schur multiplier (H2) = C23×C5

Aliases

abelianC22×C4×C5×C125
multiplicative(ℤ/13805)ˣ, (ℤ/18825)ˣ, (ℤ/20625)ˣ, (ℤ/25100)ˣ, (ℤ/27500)ˣ, (ℤ/27610)ˣ, (ℤ/37650)ˣ, (ℤ/41250)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C25C23×C52
socle, prime radicalC23×C52C2×C25
LCSG ▷ C1
UCSC1 ◁ G

540 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

10000 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC22×C5×C8×C125, C2×C42×C5×C125, C23×C4×C5×C125, …
maximal normal subgroup ofC22×C5×C8×C125, C2×C42×C5×C125, C23×C4×C5×C125, …
maximal quotient ofC22×C5×C8×C125, C2×C42×C5×C125, C23×C4×C5×C125, …
automorphism group ofC3×C11×C625, C4×C11×C625
outer automorphism group ofC3×C11×C625, C4×C11×C625

GAP

Past the SmallGroups library coverage — no SmallGroups id.