Finite Group Databasebeta

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Cayley diagram

C22×#104 #2899

order128
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 12288
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×#104

Subgroups

subgroupG/H
centerC22×C4D4
commutatorC4C23×C4
Frattini, cosocleC2×C4C24
socleC23C22⋊C4
prime radicalC22×#41C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 12 points:

(1,2);(3,4);(5,6,8,11,9,7,12,10);(5,7)(6,9)(8,10)(11,12)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×#104, S4×#104, #11712920, …
maximal subgroup of#10780, #10781, #10790, …
maximal normal subgroup of#10780, #10781, #10790, …
maximal quotient ofC22×D4⋊C8, C22×Q8⋊C8, C2×#1476, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1631);