C₇₉×#12971 #34124660

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

Aliases

direct product

C₇₉×#12971

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₇₉
prime radical	C₂³×C₇₉	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×Q₈×C₇₉ ◁ G

424 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Conjugacy classes & character table

2686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5959);