C₂×C₁₀₇×#2196 #41059507

order	27392
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₇ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×C₁₀₇×#2196

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁴×C₁₀₇
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂³×C₁₀₇ ◁ C₂²×D₄×C₁₀₇ ◁ G

1956 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 117
transitive degree	μ_t(G) = 1712

A minimal faithful permutation representation — 4 generators on 117 points:

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

Conjugacy classes & character table

4280 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 26531);