Finite Group Databasebeta

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Cayley diagram

C4×H(5,𝔽2) #3429

order128
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphismAut(G) order 73728
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC4×H(5,𝔽2)

Subgroups

subgroupG/H
centerC2×C4C24
commutatorC2C24×C4
Frattini, socle, cosocleC22C25
prime radicalC2×H(5,𝔽2)C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C22 ▷ C1

830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 12 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

derived subgroup ofSD16×#997, #72341912, #72341916, …
Fitting subgroup ofC4×#994, C4×#997, #11711015, …
maximal subgroup of#8316, #8318, #17438, …
maximal normal subgroup of#8316, #8318, #17438, …
maximal quotient ofC4×#518, C4×#520, C4×#534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2161);