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C7×Dic24 #10620599

order672
isoclinismisoclinic class of D48
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×Dic24

Splittings

HallC3×C7 ⋊ Q32
SylowC3 ⋊ C7×Q32

Subgroups

subgroupG/H
centerC2×C7D24
commutator, cosocleC3×C8C22×C7
FrattiniC8C2×D3×C7
FittingC3×C7×C16C2
socle, prime radicalC2×C3×C7D8
hypercenterC7×C16D3
hypocenterC3C7×Q32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8 ◁ C7×C16

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25)(40,41,42);(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)(13,29,21,37)(14,30,22,38)(15,31,23,39)(41,42)

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13050849, #13050854, #13050855, …
maximal normal subgroup of#13050849, #13050854, #13050855, …
maximal quotient ofC7×#4504, C7×#4503, C2×C7×Dic24

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 181);