C₈₃×#7249 #35235073

order	21248
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₈₃×#7249

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	#342
commutator	C₂²×C₄	C₄²×C₈₃
Frattini, cosocle	C₂×C₄×D₄	C₂²×C₈₃
prime radical	C₂²×D₄×C₈₃	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂²×C₄×C₈₃ ◁ C₂×C₄×D₄×C₈₃ ◁ G

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 2 generators on 107 points:

(1,2)(3,6,8,5)(4,7)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24)(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);(2,5,7,6)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)

Conjugacy classes & character table

3320 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 237);