C₂×#20274317 #27371747

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#20274317

Splittings

Fitting	C₂×C₅₃×#498 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1708

Subgroups

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#498	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#13133214	C₂²
hypercenter	C₂×#498	D₅₃
hypocenter	C₅₃	C₂×#1708
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

866 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 251 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 — 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)(56,57,58,60,59,61,62,63)(65,70)(69,77)(71,75)(72,78);(64,66,67,73,68,74,76,79)(65,69,71,72,70,77,75,78);(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)(57,60)(58,62)(61,63)(64,65)(66,72)(67,75)(68,70)(69,73)(71,76)(74,78)(77,79);(54,55)

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 157223);