Finite Group Databasebeta

← Prev (#63344) Next (#63346) →

Cayley diagram

C22×C71 #63345

order284
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C5×D3×C7 (order 420) · Inn(G) = C1 · Out(G) = C2×C5×D3×C7
homologyAbelianization (H1) = G · Schur multiplier (H2) = C2

Aliases

abelianC22×C71

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 75
transitive degreeμt(G) = 284

A minimal faithful permutation representation — 2 generators on 75 points:

(1,2)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(3,4)

Conjugacy classes & character table

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

Relationships

center ofC22⋊C4×C71, C4⋊C4×C71, C2×D4×C71, …
derived subgroup ofA4×D71, #14556042, #14556043, …
Fitting subgroup ofC2×Dic71, C22⋊D71, C22×D71, …
socle ofC2×Dic71, C2×C4×C71, C22×D71, …
maximal subgroup ofC2×Dic71, C22⋊D71, C2×C4×C71, …
maximal normal subgroup ofC2×Dic71, C22⋊D71, C2×C4×C71, …
maximal quotient ofC2×C4×C71, D4×C71, Q8×C71, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(284, 4);