#349×C₂₃₃ #28415916

order	14912
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₂₃₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

#349×C₂₃₃

Subgroups

	subgroup	G/H
center	C₄×C₂₃₃	C₂²⋊C₄
commutator	C₂²	C₂×C₈×C₂₃₃
Frattini, cosocle	C₂×C₈	C₂²×C₂₃₃
socle	C₂×C₂₃₃	C₂²⋊C₈
prime radical	C₂³×C₂₃₃	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃₃ ◁ C₂×C₈×C₂₃₃ ◁ G

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 249
transitive degree	μ_t(G) = 3728

A minimal faithful permutation representation — 2 generators on 249 points:

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

Conjugacy classes & character table

5126 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#44449745, #44449747, #44449760, …
maximal subgroup of	#1320×C₂₃₃, #1321×C₂₃₃, #1327×C₂₃₃, …
maximal normal subgroup of	#1320×C₂₃₃, #1321×C₂₃₃, #1327×C₂₃₃, …
maximal quotient of	#1314×C₂₃₃, #1322×C₂₃₃, #1330×C₂₃₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14912, 30);

#349×C233 #28415916