C₅₃×#7399 #27214911

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7399

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1343
commutator	C₂²×C₈	C₂×C₄×C₅₃
Frattini, cosocle	#408	C₂²×C₅₃
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#408 ◁ G

238 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 117 points:

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

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 387);