C₆₇×#7212 #30686463

order	17152
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#7212

Subgroups

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₇
socle	C₂²×C₆₇	#336
prime radical	C₂³×C₆₇	#97
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 91
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 2 generators on 91 points:

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

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, 200);