C₂×#20272749 #27295265

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

Aliases

direct product

C₂×#20272749

Splittings

Sylow

C₅₃ ⋊ C₂×#1356

Subgroups

	subgroup	G/H
center	C₂²	#16810362
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#97	C₂²×D₅₃
Fitting	C₂×C₅₃×#347	C₂
socle	C₂²×C₅₃	#329
prime radical	C₂³×C₅₃	#100
cosocle	#97×C₅₃	C₂³
hypercenter	C₂×#347	D₅₃
hypocenter	C₅₃	C₂×#1356
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

314 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 91 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:

(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)(56,58,61,70)(57,62,66,79)(59,71,72,76)(60,74,75,69)(63,78,65,67)(64,82,84,85)(68,73,86,80)(77,81,87,83);(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)(56,57,59,64,60,65,73,83,61,66,72,84,75,63,80,81)(58,67,68,85,69,62,76,77,70,78,86,82,74,79,71,87);(54,55)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80741);