C₇₉×#12704 #34124393

order	20224
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₉×#12704

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₇₉
prime radical	C₂²×#41×C₇₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂²×#41×C₇₉ ◁ G

962 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,4)(5,6,9,14,10,15,25,32)(7,16,18,33,19,27,23,35)(8,12,21,30,22,13,24,31)(11,26,28,20,29,34,36,17)(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);(5,7,10,19)(6,11,15,29)(8,20,22,17)(9,23,25,18)(12,16,13,27)(14,28,32,36)(21,26,24,34)(30,33,31,35);(2,4)(5,8,10,22)(6,12,15,13)(7,17,19,20)(9,21,25,24)(11,27,29,16)(14,30,32,31)(18,26,23,34)(28,35,36,33)

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, 5692);