C₂×#20272921 #27304609

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

Aliases

direct product

C₂×#20272921

Splittings

Sylow

C₅₃ ⋊ C₂×#1426

Subgroups

	subgroup	G/H
center	C₂²	#16810451
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂²×C₅₃	#367
prime radical	C₂³×C₅₃	SD₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	C₂×#1426
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

190 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 87 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(56,57,63,79,59,65,58,62,72,83,67,76,68,74,78,85,61,64,81,87,73,82,70,75,60,66,80,86,71,84,69,77);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(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)(57,62,64,75)(58,67)(59,71)(60,72)(63,78)(65,79,82,87)(66,85,83,77)(68,73)(69,81)(70,80)(74,86,84,76);(54,55)

Conjugacy classes & character table

2132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90085);