C₇₉×#13617 #34125306

order	20224
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₉×#13617

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₉
prime radical	C₂²×D₁₆×C₇₉	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂²×C₈×C₇₉ ◁ C₂²×C₁₆×C₇₉ ◁ G

330 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

4108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6605);