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Cayley diagram

C4×Q32 #2174

order128
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C22×#495725 (order 2048) · Inn(G) = D8 · Out(G) = C22×C4×D4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC4×Q32

Subgroups

subgroupG/H
centerC2×C4D8
commutatorC8C22×C4
Frattini, cosocleC2×C8C23
socle, prime radicalC22C2×D8
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

75 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22);(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#7384, #7386, C8×Q32, …
maximal normal subgroup of#7384, #7386, C8×Q32, …
maximal quotient ofC8×Q32, #12072, #12083, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 906);