C₂×#485×C₁₀₉ #28360342

order	13952
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#485×C₁₀₉

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 4 generators on 135 points:

(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(3,4)(5,8)(6,7)(9,10)(11,12,15,19)(13,16,21,25)(14,17,22,18)(20,23,26,24);(4,7)(8,10)(11,13,15,21)(12,16,19,25)(14,20,22,26)(17,23,18,24);(3,5,6,9)(4,8,7,10)(12,17)(14,22)(16,23)(18,19)(20,26)(24,25)

Conjugacy classes & character table

4142 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#9240, C₄×#485×C₁₀₉, C₁₀₉×#21583, …
maximal normal subgroup of	C₁₀₉×#9240, C₄×#485×C₁₀₉, C₁₀₉×#21583, …
maximal quotient of	C₂×C₁₀₉×#1825, C₂×C₁₀₉×#1854, C₂×C₁₀₉×#1873, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 1820);

C2×#485×C109 #28360342