D₇×#11744748 #27162440

order	12642
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

D₇×#11744748

Splittings

Fitting, socle, derived, Hall	C₇²×C₄₃ ⋊ C₂×C₃
Hall	C₇×#11744748 ⋊ C₂
Hall	C₇×D₇×C₄₃ ⋊ C₃
Hall	C₇×D₇ ⋊ C₄₃⋊C₃
Sylow	C₇² ⋊ C₂×C₄₃⋊C₃
Sylow	C₄₃ ⋊ D₇×ΓL(1,𝔽₈)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 1806

A minimal faithful permutation representation — 2 generators on 57 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,30)(18,26,31)(21,29,28)(23,32,25)(27,36,38)(33,39,37)(34,40,35)(41,43,42)(45,46)(47,48)(49,50)(52,53,54)(55,57,56);(1,2,4,8,15,17,3,6,11,18,27,26,7,13,21,30,38,36,14,23,33,40,22,31,24,34,41,43,32,12,19,28,37,42,39,20,5,9,16,25,35,29,10)(44,45,47,49,50,48,46)(51,52,53,55,54,56,57)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₇×#14268593, D₇×#14268591, D₇×#14268587, …
maximal normal subgroup of	D₇×#14268593, D₇×#14268591, D₇×#14268587, …
maximal quotient of	Dic₇×#11744748, C₂×D₇×#11744748, D₇×#14582150, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 105);

D7×#11744748 #27162440