C₆₇×#7138 #30686389

order	17152
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₈²×C₆₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₇×#7138

Subgroups

	subgroup	G/H
center	C₄×C₈×C₆₇	D₄
commutator	C₄	C₈²×C₆₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₇
socle	C₂²×C₆₇	#336
prime radical	C₂³×C₆₇	#97
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

278 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 166 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:

(9,10,12,17,13,16,20,23,14,18,21,24,11,15,19,22)(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);(1,2,3,5,4,6,7,8)(10,15,18,16)(17,22,24,23)

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, 126);