C₈₃×#11565 #35239389

order	21248
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₈₃×#11565

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₈₃
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₈₃
prime radical	C₂²×D₄×C₈₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂²×C₄×C₈₃ ◁ C₂×C₄×Q₈×C₈₃ ◁ G

412 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 107
transitive degree	μ_t(G) = 5312

A minimal faithful permutation representation — 3 generators on 107 points:

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

Conjugacy classes & character table

3071 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 4553);