C₂²×C₁₇×SD₆₄ #16891720

order	4352
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×C₁₇×SD₆₄

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₇
prime radical	C₂²×C₁₇×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ C₂²×C₁₆×C₁₇ ◁ G

502 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 53
transitive degree	μ_t(G) = 2176

A minimal faithful permutation representation — 4 generators on 53 points:

(1,2)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(6,20)(7,35)(8,18)(9,33)(10,16)(11,31)(12,14)(13,29)(15,27)(17,25)(19,23)(22,36)(24,34)(26,32)(28,30)

Conjugacy classes & character table

1292 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 26964);