#1497×C₁₉₇ #38738431

order	25216
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1497×C₁₉₇

Subgroups

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

202 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 229
transitive degree	μ_t(G) = 12608

A minimal faithful permutation representation — 3 generators on 229 points:

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

Conjugacy classes & character table

6304 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25216, 229);