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Cayley diagram

C4⋊C4×H(5,𝔽2) #7625772

order512
nilpotency class2
isoclinismisoclinic class of D4×H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C214

Aliases

direct productC4⋊C4×H(5,𝔽2)

Subgroups

subgroupG/H
center, Frattini, socle, cosocleC23C26
commutatorC22C25×C4
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

11734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 16 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72234474, #72234481, C4×D4×H(5,𝔽2), …
maximal normal subgroup of#72234474, #72234481, C4×D4×H(5,𝔽2), …
maximal quotient of#72262110, #72312130, #106×H(5,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532968);