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Cayley diagram

C2×C4⋊C4 #116

order32
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = #347042 (order 512) · Inn(G) = C22 · Out(G) = C2×B(4,𝔽2)
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4⋊C4

Subgroups

subgroupG/H
center, socle, prime radicalC23C22
commutatorC2C22×C4
Frattini, cosocleC22C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 32

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

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

derived subgroup of#7408, #7409, #7416, …
Frattini subgroup of#1318, #1319, #1324, …
Fitting subgroup ofC2×C4⋊C4×A5, C4⋊C4×SL(2,𝔽5), C2×#11751539, …
maximal subgroup of#324, #340, #343, …
maximal normal subgroup of#324, #340, #343, …
maximal quotient ofC2×#95, C4×C4⋊C4, #380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 23);