#135×C₁₀₇ #16812005

order	3424
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#135×C₁₀₇

Subgroups

	subgroup	G/H
center	C₄×C₁₀₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₁₀₇
socle	C₂×C₁₀₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₇ ◁ C₂×C₄×C₁₀₇ ◁ G

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 1712

A minimal faithful permutation representation — 3 generators on 123 points:

(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(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);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)

Conjugacy classes & character table

1498 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#359×C₁₀₇, #360×C₁₀₇, #443×C₁₀₇, …
maximal normal subgroup of	#359×C₁₀₇, #360×C₁₀₇, #443×C₁₀₇, …
maximal quotient of	#416×C₁₀₇, #427×C₁₀₇, C₄×D₈×C₁₀₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3424, 42);

#135×C107 #16812005