#413×C₂₂₇ #28408577

order	14528
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#413×C₂₂₇

Subgroups

	subgroup	G/H
center	C₄×C₂₂₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₂₂₇
Frattini, cosocle	C₂×C₄	C₂³×C₂₂₇
socle	C₂×C₂₂₇	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₂₂₇	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₂₇ ◁ C₂²×C₄×C₂₂₇ ◁ G

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 243
transitive degree	μ_t(G) = 3632

A minimal faithful permutation representation — 3 generators on 243 points:

(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(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);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)

Conjugacy classes & character table

4994 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#44397856, #44397864, #44399537, …
maximal subgroup of	#1392×C₂₂₇, #1395×C₂₂₇, #1757×C₂₂₇, …
maximal normal subgroup of	#1392×C₂₂₇, #1395×C₂₂₇, #1757×C₂₂₇, …
maximal quotient of	#1458×C₂₂₇, #1459×C₂₂₇, #1464×C₂₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14528, 94);

#413×C227 #28408577