C₄×M₁₆×C₂₃₉ #29504308

order	15296
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₃₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₄×M₁₆×C₂₃₉

Subgroups

	subgroup	G/H
center	C₄²×C₂₃₉	C₂²
commutator	C₂	C₂×C₄²×C₂₃₉
Frattini, cosocle	C₂×C₄	C₂³×C₂₃₉
socle	C₂²×C₂₃₉	C₂²×C₄
prime radical	C₂³×C₂₃₉	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃₉ ◁ G

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 251
transitive degree	μ_t(G) = 7648

A minimal faithful permutation representation — 3 generators on 251 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251);(5,6,7,10,8,9,11,12);(6,9)(10,12)

Conjugacy classes & character table

9560 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#44482386, #44482399, #44482413, …
maximal subgroup of	#1278×C₂₃₉, #1283×C₂₃₉, #1288×C₂₃₉, …
maximal normal subgroup of	#1278×C₂₃₉, #1283×C₂₃₉, #1288×C₂₃₉, …
maximal quotient of	C₈×M₁₆×C₂₃₉, #1451×C₂₃₉, #1453×C₂₃₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15296, 85);

C4×M16×C239 #29504308