C₂×#20272865 #27304338

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

Aliases

direct product

C₂×#20272865

Splittings

Fitting	C₂×C₅₃×#354 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1404

Subgroups

	subgroup	G/H
center	C₂²	#16810409
commutator	C₂×Q₈×C₅₃	C₂²×C₄
Frattini	#124	C₂²×D₅₃
Fitting	C₂×C₅₃×#354	C₂
socle	C₂²×C₅₃	AΣL(1,𝔽₁₆)
prime radical	C₂×#16810795	C₂
cosocle	#124×C₅₃	C₂³
hypercenter	C₂×#354	D₅₃
hypocenter	C₅₃	C₂×#1404
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ C₂×#354

510 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 75 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:

(3,4)(5,12)(6,15)(7,11)(8,14)(9,13)(10,17)(16,18)(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);(3,4,5,9)(6,15,16,12)(7,11,14,18)(8,17,10,13)(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);(1,2)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89814);