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C3×C112×ΓL(1,𝔽8)⋊C4 #44479004

order30492
automorphism?
homologyAbelianization (H1) = C32×C4×C112 · Schur multiplier (H2) = C3×C11

Aliases

direct productC3×C112×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C4×C112)
unambiguous semidirect productDic7⋊(C32×C112)
unambiguous semidirect product(C3×Dic7)⋊(C3×C112)
unambiguous semidirect product(C11×Dic7)⋊(C32×C11)
unambiguous semidirect product(C3×C11×Dic7)⋊(C3×C11)

Splittings

derived, SylowC7 ⋊ C32×C4×C112
HallC3×C112×ΓL(1,𝔽8) ⋊ C4
HallC112×Dic7 ⋊ C32
HallC7×C112 ⋊ C32×C4
HallC3×ΓL(1,𝔽8) ⋊ C4×C112
HallDic7 ⋊ C32×C112

Subgroups

subgroupG/H
center, hypercenterC2×C3×C112AGL(1,𝔽7)
commutator, hypocenterC7C32×C4×C112
FrattiniC2C3×C112×AGL(1,𝔽7)
Fitting, socleC2×C3×C7×C112C2×C3
prime radicalC2×C3×C112×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C32×C112
LCSG ▷ C7
UCSC1 ◁ C2×C3×C112

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 10164

Conjugacy classes & character table

5082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 227);