C₆₇×#13602 #30692853

order	17152
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#13602

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#354
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	#124	C₂³×C₆₇
socle	C₂×C₆₇	C₂×#354
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #532×C₆₇ ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 99
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 99 points:

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

Conjugacy classes & character table

2278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6590);