C₆₇×#11434 #30690685

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₆₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₇×#11434

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#99
commutator	C₂×C₄	C₂²×C₈×C₆₇
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

526 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 4288

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

(1,2,3,5,4,6,7,8)(9,10)(11,19,22,34)(12,16)(13,26,28,17)(14,18)(15,21)(20,35,23,37)(24,39,40,32)(25,33)(27,30)(29,36)(31,38)(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);(9,11,14,22)(10,15,18,30)(12,23,25,20)(13,27,28,21)(16,31,33,29)(17,34,26,19)(24,36,40,38)(32,35,39,37);(9,12,14,25)(10,16,18,33)(11,20,22,23)(13,24,28,40)(15,29,30,31)(17,32,26,39)(19,35,34,37)(21,36,27,38)

Conjugacy classes & character table

3082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 4422);