C₂×#20274022 #27365213

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#20274022

Splittings

Fitting	C₂×C₅₃×#329 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1659

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#329	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂×#16811365	C₂
hypercenter	C₂×#329	D₅₃
hypocenter	C₅₃	C₂×#1659
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#329

1150 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1150 conjugacy classes of subgroups — too many to draw. Only its 171 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) = 3392

A minimal faithful permutation representation — 4 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)(56,57)(58,63)(59,61)(60,62)(67,77)(70,75)(72,79)(73,76);(58,62)(60,63)(64,66)(65,69)(67,76)(68,74)(70,79)(71,78)(72,75)(73,77);(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)(64,65,68,71)(66,72,74,79)(67,75,77,70)(69,76,78,73);(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, 150689);