C₂×#20274013 #27365232

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20274013

Splittings

Fitting	C₂×C₅₃×#563 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1657

Subgroups

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#563	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂×#16811013	C₂
hypercenter	C₂×#563	D₅₃
hypocenter	C₅₃	C₂×#1657
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#563

1010 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 79
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 79 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)(58,63)(59,61)(60,62)(64,66,68,73)(65,70,71,79)(67,76,74,72)(69,77,75,78);(58,62)(60,63)(64,65)(66,70)(67,75)(68,71)(69,74)(72,77)(73,79)(76,78);(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)(57,60)(58,62)(61,63)(65,69)(67,74)(70,78)(71,75)(72,76)(77,79);(54,55)

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150708);