#13062015

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

derived

SL(2,𝔽₇) ⋊ C₂²

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂	C₂²×GL(3,𝔽₂)
commutator, hypocenter	SL(2,𝔽₇)	C₂²
Fitting, hypercenter	Q₈	GL(3,𝔽₂)
LCS	G ▷ SL(2,𝔽₇)
UCS	C₁ ◁ C₂ ◁ Q₈

155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,59,14,54,4,63,16,57)(2,60,7,61,5,56,3,58)(6,49,15,62,9,52,11,64)(8,55,10,53,12,51,13,50)(17,47,30,41,20,43,32,38)(18,40,23,42,21,44,19,45)(22,36,31,48,25,33,27,46)(24,35,26,34,28,39,29,37);(1,46,12,39,15,42,5,33,16,44,3,47,13,37,4,48,8,35,11,45,2,36,14,40,7,43,10,34)(6,41,9,38)(17,62,28,55,31,58,21,49,32,60,19,63,29,53,20,64,24,51,27,61,18,52,30,56,23,59,26,50)(22,57,25,54)

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209751, #72221408
maximal normal subgroup of	#72209751, #72221408
maximal quotient of	Q₈×SL(2,𝔽₇)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11304);