C₂×H(3,ℤ/4)×C₁₀₉ #28358984

order	13952
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×H(3,ℤ/4)×C₁₀₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₄²
commutator	C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#95
prime radical	C₂⁴×C₁₀₉	D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 127 points:

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

Conjugacy classes & character table

4796 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7254, C₁₀₉×#7256, C₄×H(3,ℤ/4)×C₁₀₉, …
maximal normal subgroup of	C₁₀₉×#7254, C₁₀₉×#7256, C₄×H(3,ℤ/4)×C₁₀₉, …
maximal quotient of	C₂×C₁₀₉×#1274, C₁₀₉×#7854, C₁₀₉×#7855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 462);

C2×H(3,ℤ/4)×C109 #28358984