C₂×#20272731 #27294338

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20272731

Splittings

Fitting	C₂×C₅₃×C₂²⋊C₁₆ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1349

Subgroups

	subgroup	G/H
center	C₂³	#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂×#16810859	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1349
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

498 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 3 generators on 79 points:

(1,2)(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)(56,58)(57,60)(59,63)(61,62)(65,70)(66,76)(67,68)(71,72)(73,79)(74,75)(77,78);(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)(56,57)(58,62,63,60)(59,61)(64,65,67,72,69,74,68,73)(66,75,77,71,78,70,76,79);(54,55)

Conjugacy classes & character table

2138 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79814);