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Cayley diagram

C2×C3×2−1+4 #5976

order192
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphismAut(G) order 122880
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×C3×2−1+4

Subgroups

subgroupG/H
center, socleC22×C3C24
commutator, Frattini, cosocleC2C25×C3
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 21 points:

(1,2)(3,4,5);(7,11)(9,17)(12,14)(15,21);(6,7,10,11)(8,14,16,12)(9,13,17,18)(15,20,21,19);(6,8,10,16)(7,12,11,14)(9,15,17,21)(13,19,18,20);(1,2)(6,9)(7,13)(8,15)(10,17)(11,18)(12,19)(14,20)(16,21)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72143, #72144, #84111, …
maximal subgroup of#72143, #72144, C3×#1793, …
maximal normal subgroup of#72143, #72144, C3×#1793, …
maximal quotient ofC2×C3×#519, C2×C3×#520, C3×C4×2−1+4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1535);