C₇₉×#7216 #34118905

order	20224
nilpotency class	4
isoclinism	isoclinic class of #7213
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₉×#7216

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#342
commutator	C₂²×C₄	C₄²×C₇₉
Frattini, cosocle	C₂×#117	C₂²×C₇₉
prime radical	C₂⁴×C₇₉	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×#117×C₇₉ ◁ G

276 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 119
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 generators on 119 points:

(1,2)(3,5)(4,7)(6,8)(9,10,12,17,14,16,23,29)(11,19,20,31,22,15,26,27)(13,18,24,32,25,30,36,38)(21,33,34,40,35,28,37,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,5,7,6)(9,11,13,21,14,22,25,35)(10,15,18,28,16,19,30,33)(12,20,24,34,23,26,36,37)(17,31,32,40,29,27,38,39)

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, 204);