C₇₉×#8188 #34119877

order	20224
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#8188

Subgroups

	subgroup	G/H
center	C₄²×C₇₉	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₇₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

412 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 10112

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

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

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, 1176);