C₁₀₇×#7021 #41032985

order	27392
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₀₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₇×#7021

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₇	#95
commutator	C₂×C₄	C₄×C₈×C₁₀₇
Frattini, cosocle	C₂²×C₄²	C₂²×C₁₀₇
prime radical	C₂⁴×C₁₀₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂²×C₄²×C₁₀₇ ◁ G

380 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 131
transitive degree	μ_t(G) = 13696

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

(1,2)(3,5)(4,7)(6,8)(9,10)(11,15,16,13)(12,14)(17,18,20,23,22,24,21,19)(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);(2,5,7,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,24,22,23,20,18)

Conjugacy classes & character table

6848 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27392, 9);