C₈₃×#7188 #35235012

order	21248
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₈₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₈₃×#7188

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈₃	#99
commutator	C₂³	C₄×C₈×C₈₃
Frattini, cosocle	C₂×#117	C₂²×C₈₃
socle	C₂²×C₈₃	#342
prime radical	C₂⁴×C₈₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₃ ◁ C₂²×C₄×C₈₃ ◁ C₂×#117×C₈₃ ◁ G

322 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 123
transitive degree	μ_t(G) = 5312

A minimal faithful permutation representation — 2 generators on 123 points:

(1,2,3,5,4,6,7,8)(9,10)(11,19)(12,23,24,16)(13,17)(14,18)(15,20,27,33)(21,31)(22,32)(25,36,37,29)(26,30)(28,34,38,40)(35,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(9,11,13,21,14,22,26,35)(10,15,17,28,18,27,30,38)(12,20,25,34,24,33,37,40)(16,19,29,31,23,32,36,39)

Conjugacy classes & character table

4316 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 176);