C₂×#2203×C₁₃₉ #48028159

order	35584
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#2203×C₁₃₉

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₁₃₉
prime radical	C₂×#454×C₁₃₉	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂³×C₁₃₉ ◁ C₂²×Q₈×C₁₃₉ ◁ G

1092 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 173
transitive degree	μ_t(G) = 8896

A minimal faithful permutation representation — 4 generators on 173 points:

(1,2)(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);(3,4)(5,14)(6,18)(7,10)(8,13)(9,15)(11,21)(12,23)(16,31)(17,29)(19,27)(20,28)(22,32)(24,30)(25,33)(26,34);(3,5,8,17)(4,9,13,27)(6,19,21,15)(7,22,23,16)(10,25,12,26)(11,29,18,14)(20,33,24,34)(28,31,30,32);(4,10)(7,23)(9,25)(11,30)(12,13)(14,31)(16,22)(18,28)(20,24)(26,27)(29,32)(33,34)

Conjugacy classes & character table

4726 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35584, 26551);