C₆₇×#7252 #30686503

order	17152
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₇×#7252

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	#99
commutator	C₂²×C₄	C₄²×C₆₇
Frattini, cosocle	C₂×#117	C₂²×C₆₇
socle	C₂²×C₆₇	#342
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

344 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 2 generators on 107 points:

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

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, 240);