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ΓL(1,𝔽8)×Dic119 #23756155

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

Aliases

direct productΓL(1,𝔽8)×Dic119

Splittings

derived, HallC72×C17 ⋊ C3×C4
HallC7×Dic119 ⋊ C3
HallC7×C17×ΓL(1,𝔽8) ⋊ C4
HallC7×ΓL(1,𝔽8) ⋊ Dic17
SylowC72 ⋊ C3×Dic17
SylowC17 ⋊ ΓL(1,𝔽8)×Dic7

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D119
commutator, hypocenterC72×C17C3×C4
Fitting, socle, cosocleC2×C72×C17C2×C3
prime radicalC2×C7×C17×ΓL(1,𝔽8)C2
LCSG ▷ C72×C17
UCSC1 ◁ C2

52 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

610 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×ΓL(1,𝔽8)⋊Dic17, C3×ΓL(1,𝔽8)×Dic119, ΓL(1,𝔽8)×Dic357
maximal normal subgroup ofΓL(1,𝔽8)×ΓL(1,𝔽8)⋊Dic17, C3×ΓL(1,𝔽8)×Dic119
maximal quotient ofC7⋊C9×Dic119, C3×ΓL(1,𝔽8)×Dic119, ΓL(1,𝔽8)×Dic357

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 58);