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Cayley diagram

C22×#108 #2914

order128
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 24576
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC22×#108

Subgroups

subgroupG/H
centerC22×C4D4
commutatorC4C23×C4
Frattini, cosocleC2×C4C24
socleC23C4⋊C4
prime radicalC24Q8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×#108, #11711548, #11711552, …
maximal subgroup of#10859, #10860, #10864, …
maximal normal subgroup of#10859, #10860, #10864, …
maximal quotient ofC22×#334, C22×#335, C2×#1564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1646);