C₁₀₇×#2875 #28333558

order	13696
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₁₀₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₀₇×#2875

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₇	C₂⁴
commutator	C₂	C₂²×C₄²×C₁₀₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₇
socle	C₂²×C₁₀₇	C₂³×C₄
prime radical	C₂×#41×C₁₀₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ G

556 conjugacy classes of subgroups; only the 504 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 504 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) = 6848

A minimal faithful permutation representation — 4 generators on 127 points:

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

Conjugacy classes & character table

7276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₇×#9311, C₁₀₇×#9312, C₁₀₇×#12609, …
maximal normal subgroup of	C₁₀₇×#9311, C₁₀₇×#9312, C₁₀₇×#12609, …
maximal quotient of	C₁₀₇×#17316, C₁₀₇×#17320, C₁₀₇×#17329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 1607);