C₁₀₇×#33567 #41059531

order	27392
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₁₀₇×#33567

Subgroups

	subgroup	G/H
center	C₄×C₁₀₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₁₀₇
socle	C₂×C₁₀₇	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₇ ◁ C₂×C₄×C₁₀₇ ◁ C₂×#41×C₁₀₇ ◁ G

1184 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 123
transitive degree	μ_t(G) = 1712

A minimal faithful permutation representation — 4 generators on 123 points:

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

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, 26555);