C₄×M₁₆×C₁₉₉ #27163554

order	12736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₉₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₄×M₁₆×C₁₉₉

Subgroups

	subgroup	G/H
center	C₄²×C₁₉₉	C₂²
commutator	C₂	C₂×C₄²×C₁₉₉
Frattini, cosocle	C₂×C₄	C₂³×C₁₉₉
socle	C₂²×C₁₉₉	C₂²×C₄
prime radical	C₂³×C₁₉₉	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₉₉ ◁ G

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 211
transitive degree	μ_t(G) = 6368

A minimal faithful permutation representation — 3 generators on 211 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211);(5,6,7,10,8,9,11,12);(6,9)(10,12)

Conjugacy classes & character table

7960 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#38766688, #38766701, #38766715, …
maximal subgroup of	#1278×C₁₉₉, #1283×C₁₉₉, #1288×C₁₉₉, …
maximal normal subgroup of	#1278×C₁₉₉, #1283×C₁₉₉, #1288×C₁₉₉, …
maximal quotient of	C₈×M₁₆×C₁₉₉, #1451×C₁₉₉, #1453×C₁₉₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12736, 85);

C4×M16×C199 #27163554