C₂²×C₄×D₄×C₁₀₇ #28334105

order	13696
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₁₀₇ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×C₄×D₄×C₁₀₇

Subgroups

	subgroup	G/H
center	C₂³×C₄×C₁₀₇	C₂²
commutator	C₂	C₂⁴×C₄×C₁₀₇
Frattini, cosocle	C₂²	C₂⁵×C₁₀₇
socle	C₂⁴×C₁₀₇	C₂³
prime radical	C₂³×D₄×C₁₀₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₀₇ ◁ G

2528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 6848

A minimal faithful permutation representation — 5 generators on 119 points:

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

Conjugacy classes & character table

8560 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₇×#8301, C₁₀₇×#8302, C₁₀₇×#8310, …
maximal normal subgroup of	C₁₀₇×#8301, C₁₀₇×#8302, C₁₀₇×#8310, …
maximal quotient of	C₂²×C₄×C₂²⋊C₄×C₁₀₇, C₂²×C₄×C₄⋊C₄×C₁₀₇, C₂×C₄²×D₄×C₁₀₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 2154);

C22×C4×D4×C107 #28334105