C₁₀₇×#1324 #28332007

order	13696
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₀₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₇×#1324

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#99
commutator	C₂×C₄	C₂×C₈×C₁₀₇
Frattini, cosocle	C₂×C₄⋊C₄	C₂²×C₁₀₇
prime radical	C₂³×C₁₀₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂³×C₁₀₇ ◁ C₂×C₄⋊C₄×C₁₀₇ ◁ G

118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 3424

A minimal faithful permutation representation — 2 generators on 131 points:

(1,2,3,5,4,6,7,8)(9,10)(11,17,19,23)(12,20,21,15)(13,16)(14,18)(22,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(9,11)(10,14,16,22)(12,18)(13,19)(15,17,20,23)(21,24)

Conjugacy classes & character table

2782 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₇×#7352, C₁₀₇×#7353, C₁₀₇×#7359, …
maximal normal subgroup of	C₁₀₇×#7352, C₁₀₇×#7353, C₁₀₇×#7359, …
maximal quotient of	C₁₀₇×#7062, C₁₀₇×#7065, C₁₀₇×#7122, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 56);

C107×#1324 #28332007