C₆₇×#12862 #30692113

order	17152
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#12862

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂⁴×C₆₇ ◁ C₂³×Q₈×C₆₇ ◁ G

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

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 107 points:

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

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, 5850);