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Cayley diagram

C4×2−1+4 #3430

order128
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphismAut(G) order 122880
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC4×2−1+4

Subgroups

subgroupG/H
centerC2×C4C24
commutatorC2C24×C4
Frattini, socle, cosocleC22C25
prime radicalC2×2−1+4C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C22 ▷ C1

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4×2−1+4⋊C3, C4×2−1+4⋊C5, #11711016, …
maximal subgroup of#8319, #17439, #17454, …
maximal normal subgroup of#8319, #17439, #17454, …
maximal quotient ofC4×#520, C4×#519, C4×#536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2162);