C₂²×SD₆₄×C₆₇ #30713227

order	17152
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×SD₆₄×C₆₇

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₆₇
prime radical	C₂²×D₁₆×C₆₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ C₂²×C₁₆×C₆₇ ◁ 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) = 103
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 4 generators on 103 points:

(1,2)(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);(3,4);(5,6,10,15)(7,11,19,26)(8,20,22,36)(9,23,21,14)(12,27,29,16)(13,30,31,17)(18,32,33,25)(24,35,28,34);(6,11)(7,16)(8,17)(9,21)(12,13)(14,32)(15,26)(18,35)(19,27)(20,24)(22,30)(23,25)(28,36)(29,31)(33,34)

Conjugacy classes & character table

5092 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 26964);