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C17×ΓL(1,𝔽8)×Dic7 #23756154

order9996
automorphism?
homologyAbelianization (H1) = C3×C4×C17 · Schur multiplier (H2) = C1

Aliases

direct productC17×ΓL(1,𝔽8)×Dic7

Splittings

derived, SylowC72 ⋊ C3×C4×C17
HallC7×C17×Dic7 ⋊ C3
HallC7×C17×ΓL(1,𝔽8) ⋊ C4
HallC72×C17 ⋊ C3×C4
HallC7×Dic7 ⋊ C3×C17
HallC7×ΓL(1,𝔽8) ⋊ C4×C17

Subgroups

subgroupG/H
center, hypercenterC2×C17D7×ΓL(1,𝔽8)
commutator, hypocenterC72C3×C4×C17
FrattiniC2D7×C17×ΓL(1,𝔽8)
Fitting, socleC2×C72×C17C2×C3
prime radicalC2×C7×C17×ΓL(1,𝔽8)C2
cosocleC2×C72C2×C3×C17
LCSG ▷ C72
UCSC1 ◁ C2×C17

52 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

850 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC17×ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4, C3×C17×ΓL(1,𝔽8)×Dic7, C17×ΓL(1,𝔽8)×Dic21
maximal normal subgroup ofC17×ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4, C3×C17×ΓL(1,𝔽8)×Dic7
maximal quotient ofC17×Dic7×C7⋊C9, C3×C17×ΓL(1,𝔽8)×Dic7, C17×ΓL(1,𝔽8)×Dic21

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 57);