C₇₉×#12724 #34124413

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₉×#12724

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1782
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×#41	C₂³×C₇₉
prime radical	C₂×#41×C₇₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₄×#41×C₇₉ ◁ G

670 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 138 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:

(1,2,6,12)(3,14,17,29)(4,10,5,11)(7,13,20,28)(8,16,25,15)(9,19,27,22)(18,31,23,24)(21,30,32,26)(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);(1,3,7,18)(2,8,13,26)(4,15,19,32)(5,21,22,16)(6,23,20,17)(9,24,11,29)(10,14,27,31)(12,25,28,30);(2,9,13,11)(4,19)(6,20)(8,24,26,29)(10,28,27,12)(14,30,31,25)(15,32)(17,23)

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, 5712);