C₆₇×Q₈⋊C₃₂ #30686643

order	17152
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₆₇×Q₈⋊C₃₂
unambiguous semidirect product	Q₈⋊(C₃₂×C₆₇)

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₆₇	D₄
commutator	C₄	C₂×C₃₂×C₆₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₆₇
socle, prime radical	C₂²×C₆₇	C₂²⋊C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

118 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 17152

A minimal faithful permutation representation — 2 generators on 107 points:

(2,5)(3,7)(6,8)(9,10,11,15,12,16,19,25,13,17,20,26,22,28,31,35,14,18,21,27,23,29,32,36,24,30,33,37,34,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,2,4,6)(3,8,7,5)

Conjugacy classes & character table

7504 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 380);