C₇₉×#12045 #34123734

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#12045

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂×D₈
commutator	C₈	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₄×D₈
prime radical	C₂×D₁₆×C₇₉	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₈²×C₇₉ ◁ G

254 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 103
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 103 points:

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

Conjugacy classes & character table

5056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5033);