#2198×C₂₃₉ #44480968

order	30592
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₃₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#2198×C₂₃₉

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₃₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂³×C₂₃₉
prime radical	#453×C₂₃₉	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃₉ ◁ C₂²×C₂₃₉ ◁ C₂×D₄×C₂₃₉ ◁ G

246 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 255
transitive degree	μ_t(G) = 3824

A minimal faithful permutation representation — 3 generators on 255 points:

(1,2)(3,11)(4,8)(5,9)(6,10)(7,12)(13,15)(14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255);(1,3,6,4)(2,7,10,15)(5,12,14,13)(8,9,11,16);(1,4,6,3)(2,8,10,11)(5,12,14,13)(7,9,15,16)

Conjugacy classes & character table

4780 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30592, 930);