C₈₃×D₄⋊C₃₂ #35235203

order	21248
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂×C₈₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₈₃×D₄⋊C₃₂
unambiguous semidirect product	D₄⋊(C₃₂×C₈₃)

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₈₃	D₄
commutator	C₄	C₂×C₃₂×C₈₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₃
socle	C₂²×C₈₃	C₂²⋊C₁₆
prime radical	C₂×D₄×C₈₃	C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₈₃ ◁ C₄×C₁₆×C₈₃ ◁ G

138 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 10624

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

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

Conjugacy classes & character table

9296 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 367);