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Cayley diagram

C22×#100 #2885

order128
nilpotency class3
isoclinismisoclinic class of #99
automorphismAut(G) order 49152
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×#100

Subgroups

subgroupG/H
center, socleC23C22⋊C4
commutatorC22C23×C4
Frattini, cosocleC2×C4C24
prime radicalC23×D4C2
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

496 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);(5,6,7,10,8,9,11,12);(6,9)(7,11)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×#100, #11711610, #11711611, …
maximal subgroup of#10740, #10749, #10762, …
maximal normal subgroup of#10740, #10749, #10762, …
maximal quotient ofC22×#323, C2×#1460, C2×#1462, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1617);