C₂×#20272929 #27305137

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

Aliases

direct product

C₂×#20272929

Splittings

Fitting	C₂×C₅₃×Q₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×SD₁₂₈

Subgroups

	subgroup	G/H
center	C₂²	#16810458
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×Q₆₄	C₂
socle	C₂²×C₅₃	D₃₂
prime radical	C₂×D₁₆₉₆	C₂
hypercenter	C₂×Q₆₄	D₅₃
hypocenter	C₅₃	C₂×SD₁₂₈
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×Q₆₄

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

normal subgroups only

This group has 134 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)(66,94)(67,96)(68,85)(71,102)(72,105)(73,86)(74,82)(76,110)(77,90)(78,83)(80,93)(81,84)(87,107)(88,106)(89,92)(91,112)(95,119)(97,98)(99,103)(100,101)(104,118)(108,115)(109,114)(111,117)(113,116);(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)(56,57,62,68)(58,63,73,85)(59,74,77,108)(60,78,80,111)(61,81,79,67)(64,86,89,69)(65,90,92,70)(66,93,95,75)(71,94,104,119)(72,96,103,84)(76,109,100,107)(82,113,115,98)(83,116,117,97)(87,118,114,102)(88,99,112,105)(91,101,106,110);(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, 90613);