Finite Group Databasebeta

← Prev (#39) Next (#41) →

Cayley diagram

C2×Q8 #40

order16
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = #1020⋊C2 (order 192) · Inn(G) = C22 · Out(G) = C2×S4
holomorphG ⋊ Aut(G) = #72210418
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×Q8

Subgroups

subgroupG/H
center, socle, prime radicalC22C22
commutator, Frattini, cosocleC2C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C2 ▷ C1

19 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 3 generators on 10 points:

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

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9C10
size1122211222
order1244422444
χ11111111111
χ2111-1-1111-1-1
χ311-11-111-11-1
χ411-1-1111-1-11
χ52-20002-2000
χ611111-1-1-1-1-1
χ7111-1-1-1-1-111
χ811-11-1-1-11-11
χ911-1-11-1-111-1
χ102-2000-22000

Relationships

derived subgroup of#1404, #1405, #2202, …
Frattini subgroup of#355, #356, C2×#355, …
Fitting subgroup ofC2×SL(2,𝔽3), #859, C2×#226, …
maximal subgroup of#101, #103, C4×Q8, …
maximal normal subgroup of#101, #103, C4×Q8, …
maximal quotient ofC2×C4⋊C4, C4×Q8, C22⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16, 12);