M₁₆×#11715906 #27158844

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

M₁₆×#11715906

Splittings

Fitting	M₁₆×C₉₇ ⋊ C₈
Sylow	C₉₇ ⋊ C₈×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂²×#11715906
commutator, socle	C₂×C₉₇	C₂×C₄×C₈
Fitting	M₁₆×C₉₇	C₈
prime radical	C₂²×D₉₇	C₄²
cosocle	C₄×#87604	C₂³
hypercenter	M₁₆	#11715906
hypocenter	C₉₇	C₈×M₁₆
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ M₁₆

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 105
transitive degree	μ_t(G) = 776

A minimal faithful permutation representation — 3 generators on 105 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21618);