C₁₀₉×#2222 #28359476

order	13952
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#2222

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
prime radical	#41×C₁₀₉	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ #131×C₁₀₉ ◁ G

140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 173
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 173 points:

(1,2,8,14)(3,15,21,43)(4,22,25,39)(5,26,28,37)(6,13,7,12)(9,30,34,18)(10,29,36,24)(11,27,38,16)(17,46,49,63)(19,42,20,41)(23,54,55,35)(31,58,59,64)(32,50,33,44)(40,51,62,48)(45,53,56,61)(47,52,57,60)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(1,3,16,44,7,20,28,30,8,21,27,50,6,19,5,18)(2,9,26,42,13,33,38,43,14,34,37,41,12,32,11,15)(4,17,45,64,24,48,55,57,25,49,56,58,29,51,23,47)(10,31,53,46,22,52,54,62,36,59,61,63,39,60,35,40);(1,4)(2,10)(3,17)(5,23)(6,29)(7,24)(8,25)(9,31)(11,35)(12,39)(13,22)(14,36)(15,40)(16,45)(18,47)(19,51)(20,48)(21,49)(26,53)(27,56)(28,55)(30,57)(32,60)(33,52)(34,59)(37,61)(38,54)(41,63)(42,46)(43,62)(44,64)(50,58)

Conjugacy classes & character table

2834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#13671, C₁₀₉×#13672, C₁₀₉×#13701, …
maximal normal subgroup of	C₁₀₉×#13671, C₁₀₉×#13672, C₁₀₉×#13701, …
maximal quotient of	C₁₀₉×#12431, C₁₀₉×#12434, C₁₀₉×#12440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 954);