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Cayley diagram

C2×Q8×C4⋊C4 #34112

order256
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) order 6291456
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×Q8×C4⋊C4

Subgroups

subgroupG/H
center, socle, prime radicalC24C24
commutatorC22C24×C4
Frattini, cosocleC23C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

1623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×C4⋊C4×SL(2,𝔽3)
maximal subgroup of#97587, #105992, #106729, …
maximal normal subgroup of#97587, #105992, #106729, …
maximal quotient ofC2×C4⋊C42, C2×Q8×#95, C2×#13818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27100);