C₁₀₇×#1296 #28331979

order	13696
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₇×#1296

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₇	C₂²⋊C₄
commutator	C₂×C₄	C₄²×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂²×C₁₀₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

188 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 127
transitive degree	μ_t(G) = 13696

A minimal faithful permutation representation — 2 generators on 127 points:

(1,2,3,4)(6,10)(7,8)(11,12)(13,14)(15,17)(16,19)(18,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(5,6,8,11,9,12,7,10)(14,17)(15,20)(18,19)

Conjugacy classes & character table

4066 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₇×#7417, C₁₀₇×#7443, C₂×C₁₀₇×#1296, …
maximal normal subgroup of	C₁₀₇×#7417, C₁₀₇×#7443, C₂×C₁₀₇×#1296, …
maximal quotient of	C₁₀₇×#7025, C₁₀₇×#7031, C₁₀₇×#7039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 28);