C₂×#20272927 #27305134

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

Aliases

direct product

C₂×#20272927

Splittings

Fitting	C₂×C₅₃×D₃₂ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×D₆₄

Subgroups

	subgroup	G/H
center	C₂²	#16810458
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×D₃₂	C₂
socle	C₂²×C₅₃	D₃₂
hypercenter	C₂×D₃₂	D₅₃
hypocenter	C₅₃	C₂×D₆₄
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×D₃₂

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 119
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 119 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)(57,63)(58,69)(59,70)(60,75)(61,79)(64,65)(66,91)(67,93)(68,85)(71,101)(72,104)(73,97)(74,82)(76,110)(77,100)(78,83)(80,108)(81,84)(86,106)(87,105)(88,90)(89,112)(92,116)(94,95)(96,102)(98,99)(103,119)(107,114)(109,113)(111,115)(117,118);(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)(59,74)(60,78)(61,81)(62,68)(64,69)(65,70)(66,75)(67,79)(71,91)(72,93)(73,85)(76,109)(77,107)(80,111)(82,95)(83,94)(84,102)(86,101)(87,104)(88,97)(89,110)(90,100)(92,108)(96,112)(98,106)(99,105)(103,116)(113,119)(114,118)(115,117);(54,55)

Conjugacy classes & character table

1058 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90610);