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Cayley diagram

C23×#101 #60072

order256
nilpotency class3
isoclinismisoclinic class of #99
automorphismAut(G) order 11010048
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC23×#101

Subgroups

subgroupG/H
center, socleC24C22⋊C4
commutatorC22C24×C4
Frattini, cosocleC2×C4C25
prime radicalC25C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

2495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(3,4);(5,6);(7,8,10,14,11,13,19,21)(9,15,16,22,17,12,18,20);(7,9,11,17)(8,12,13,15)(10,18,19,16)(14,22,21,20)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×A4×#101, #101×AGL(1,𝔽8)
maximal subgroup of#123522, #123529, #123541, …
maximal normal subgroup of#123522, #123529, #123541, …
maximal quotient ofC23×#324, C22×#1461, C22×#1463, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53060);