C₄×Q₃₂×C₁₀₇ #28332857

order	13696
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄×Q₃₂×C₁₀₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₇	D₈
commutator	C₈	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₇
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₄²×C₁₀₇ ◁ C₄×C₈×C₁₀₇ ◁ G

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 13696

A minimal faithful permutation representation — 3 generators on 143 points:

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

Conjugacy classes & character table

4708 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₇×#7384, C₁₀₇×#7386, C₈×Q₃₂×C₁₀₇, …
maximal normal subgroup of	C₁₀₇×#7384, C₁₀₇×#7386, C₈×Q₃₂×C₁₀₇, …
maximal quotient of	C₈×Q₃₂×C₁₀₇, C₁₀₇×#12072, C₁₀₇×#12083, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 906);

C4×Q32×C107 #28332857