C₁₀₇×#1596 #28332279

order	13696
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₇×#1596

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₇
prime radical	#392×C₁₀₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ G

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 123
transitive degree	μ_t(G) = 1712

A minimal faithful permutation representation — 3 generators on 123 points:

(2,6)(3,9)(7,13)(8,10)(11,15)(14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,2,4,8)(3,10,9,6)(5,7,12,14)(11,16,15,13);(3,11)(6,13)(9,15)(10,16)

Conjugacy classes & character table

2461 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₇×#12202, C₁₀₇×#12204, C₁₀₇×#12205, …
maximal normal subgroup of	C₁₀₇×#12202, C₁₀₇×#12204, C₁₀₇×#12205, …
maximal quotient of	C₁₀₇×#7735, C₁₀₇×#7738, C₁₀₇×#7743, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 328);