#1360×C₁₇₃ #36349145

order	22144
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₇₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

#1360×C₁₇₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₇₃	#102
commutator	C₂×C₈	C₂×C₄×C₁₇₃
Frattini, cosocle	C₄×C₈	C₂²×C₁₇₃
prime radical	C₂×D₈×C₁₇₃	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇₃ ◁ C₄²×C₁₇₃ ◁ C₄×C₈×C₁₇₃ ◁ G

102 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 197
transitive degree	μ_t(G) = 11072

A minimal faithful permutation representation — 2 generators on 197 points:

(1,2,3,6,4,5,7,8)(10,15,16,23)(11,19)(12,13)(17,20,18,24)(21,22)(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);(2,5)(3,7)(9,10)(11,20)(12,18)(13,17)(14,16)(15,21)(19,23)(22,24)

Conjugacy classes & character table

5017 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22144, 92);