C₇₉×#12975 #34124664

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#12975

Subgroups

	subgroup	G/H
center	C₄×C₇₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×#41	C₂³×C₇₉
socle	C₂×C₇₉	#1867
prime radical	#532×C₇₉	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂×#41×C₇₉ ◁ G

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

normal subgroups only

This group has 416 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) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 points:

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

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5963);