C₂×#20272860 #27304307

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20272860

Splittings

Sylow

C₅₃ ⋊ C₂×#1402

Subgroups

	subgroup	G/H
center	C₂²	#16810410
commutator	C₂×D₄×C₅₃	C₂²×C₄
Frattini	#127	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#453	C₂
socle	C₂²×C₅₃	AΣL(1,𝔽₁₆)
cosocle	#127×C₅₃	C₂³
hypercenter	C₂×#453	D₅₃
hypocenter	C₅₃	C₂×#1402
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#127 ◁ C₂×#453

564 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 71 points:

(3,5)(4,8)(6,15)(7,14)(9,12)(10,18)(11,17)(13,16)(19,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21);(3,4)(5,12,13,8)(6,10,16,9)(7,11)(14,18,15,17)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71);(1,2)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89783);