C₂×#441×C₁₆₇ #36321301

order	21376
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#441×C₁₆₇

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₆₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₁₆₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₆₇ ◁ C₂×C₄²×C₁₆₇ ◁ G

480 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 189
transitive degree	μ_t(G) = 21376

A minimal faithful permutation representation — 4 generators on 189 points:

(1,2)(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);(3,4,6,5)(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21);(7,9,11,17)(8,12,13,21)(10,19,20,16)(14,22,18,15);(3,5,6,4)(8,13)(12,21)(14,18)(15,22)

Conjugacy classes & character table

7348 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21376, 1673);