C₅₃×#12112 #27219624

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#12112

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#408 ◁ G

390 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 125
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 125 points:

(5,8)(9,10,16,22)(11,23,27,51)(12,28,31,48)(13,32,36,46)(14,21,15,20)(17,39,42,25)(18,37,44,30)(19,34,47,35)(24,52,56,69)(26,57,38,45)(29,60,63,43)(33,49,64,41)(40,61,68,72)(50,55,71,54)(53,58,66,67)(59,65,70,62)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,2)(3,5)(4,6)(7,8)(9,11,16,27)(10,17,22,42)(12,24,31,56)(13,33,36,64)(14,26,15,38)(18,40,44,68)(19,45,47,57)(20,41,21,49)(23,46,51,32)(25,34,39,35)(28,58,48,67)(29,61,63,72)(30,54,37,55)(43,69,60,52)(50,70,71,59)(53,65,66,62);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,24)(13,29)(14,37)(15,30)(16,31)(17,40)(19,43)(20,48)(21,28)(22,44)(23,50)(25,53)(26,55)(27,56)(32,59)(33,61)(34,65)(35,62)(36,63)(38,54)(39,66)(41,67)(42,68)(45,69)(46,70)(47,60)(49,58)(51,71)(52,57)(64,72)

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5100);