C₁₀₇×#2170 #28332853

order	13696
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₇×#2170

Subgroups

	subgroup	G/H
center	C₁₆×C₁₀₇	D₄
commutator	C₄	C₂²×C₈×C₁₀₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₇
socle	C₂×C₁₀₇	C₈×D₄
prime radical	#135×C₁₀₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₀₇ ◁ C₂×C₁₆×C₁₀₇ ◁ G

146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 139
transitive degree	μ_t(G) = 3424

A minimal faithful permutation representation — 3 generators on 139 points:

(1,2,6,10,5,11,20,28,7,12,22,29,21,9,25,27)(3,13,15,18,16,24,30,19,17,26,31,32,14,8,23,4)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,3)(2,8)(4,10)(5,16)(6,15)(7,17)(9,26)(11,13)(12,24)(14,21)(18,28)(19,29)(20,30)(22,31)(23,25)(27,32);(1,4,6,13,5,18,20,24,7,19,22,26,21,32,25,8)(2,3,10,15,11,16,28,30,12,17,29,31,9,14,27,23)

Conjugacy classes & character table

5992 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₇×#7381, C₁₀₇×#7382, C₁₀₇×#12046, …
maximal normal subgroup of	C₁₀₇×#7381, C₁₀₇×#7382, C₁₀₇×#12046, …
maximal quotient of	C₁₆×D₈×C₁₀₇, C₁₀₇×#12039, C₁₆×SD₁₆×C₁₀₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 902);