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Cayley diagram

C4×D4×Q8 #34161

order256
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) order 393216
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC4×D4×Q8

Subgroups

subgroupG/H
centerC22×C4C24
commutatorC22C24×C4
Frattini, socle, cosocleC23C25
prime radicalC22×D4C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

1574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 16 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4×D4×SL(2,𝔽3)
maximal subgroup of#97662, C4×#1627, C4×#1637, …
maximal normal subgroup of#97662, C4×#1627, C4×#1637, …
maximal quotient ofC4×D4×C4⋊C4, C4×Q8×C22⋊C4, C4×Q8×C4⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27149);