#1296×C₁₇₉ #37504518

order	22912
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#1296×C₁₇₉

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇₉	C₂²⋊C₄
commutator	C₂×C₄	C₄²×C₁₇₉
Frattini, cosocle	C₂×C₄²	C₂²×C₁₇₉
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇₉ ◁ C₂×C₄²×C₁₇₉ ◁ G

188 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 199
transitive degree	μ_t(G) = 22912

A minimal faithful permutation representation — 2 generators on 199 points:

(1,2,3,4)(6,10)(7,8)(11,12)(13,14)(15,17)(16,19)(18,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199);(5,6,8,11,9,12,7,10)(14,17)(15,20)(18,19)

Conjugacy classes & character table

6802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22912, 28);