C₂×#332×C₂₃₉ #44480309

order	30592
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₃₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#332×C₂₃₉

Subgroups

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

256 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 257
transitive degree	μ_t(G) = 30592

A minimal faithful permutation representation — 3 generators on 257 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257);(4,7)(5,9)(8,10)(11,12,14,16,15,18,13,17);(3,4,6,8)(5,10,9,7)(12,16)(13,14)(17,18)

Conjugacy classes & character table

9082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30592, 271);