Finite Group Databasebeta

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Cayley diagram

C2×#458 #3024

order128
nilpotency class3
isoclinismisoclinic class of B(4,𝔽2)
automorphismAut(G) = #72240913 (order 6144) · Inn(G) = #120 · Out(G) = D4×S4
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#458

Subgroups

subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC23C24
prime radicalC2×H(5,𝔽2)C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C2 ▷ C1

397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#458⋊C3, #11711581, #11711586, …
maximal subgroup of#13042, #13044, #13344, …
maximal normal subgroup of#13042, #13044, #13344, …
maximal quotient ofC2×#1607, C2×#1613, C2×#1608, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1756);