#1579×C₁₉₇ #38738513

order	25216
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1579×C₁₉₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉₇	C₂×D₄
commutator	C₄	C₂²×C₈×C₁₉₇
Frattini, cosocle	C₄²	C₂³×C₁₉₇
socle, prime radical	C₂²×C₁₉₇	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉₇ ◁ C₄×C₈×C₁₉₇ ◁ G

160 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 221
transitive degree	μ_t(G) = 25216

A minimal faithful permutation representation — 3 generators on 221 points:

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

Conjugacy classes & character table

8668 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25216, 311);