C₂×#20272710 #27294251

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

Aliases

direct product

C₂×#20272710

Splittings

Sylow

C₅₃ ⋊ C₂×#1342

Subgroups

	subgroup	G/H
center	C₂²	#16810358
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂²×C₅₃	#327
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	C₂×#1342
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#349

428 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 87 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)(56,58,62,71)(57,63,66,80)(59,73,76,85)(60,70,61,69)(64,81,83,87)(65,78,77,79)(67,82,84,72)(68,75,86,74);(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)(56,57,60,65,62,66,61,77)(58,67,69,81,71,84,70,87)(59,72,74,64,76,82,75,83)(63,73,78,86,80,85,79,68);(54,55)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79727);