C₄×C₆₇×C₁₂₁ #44569812

order	32428
nilpotency class	1
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₂₄₂₈
abelian	C₄×C₆₇×C₁₂₁
multiplicative	(ℤ/32429)ˣ, (ℤ/64858)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₁	C₂×C₁₁×C₆₇
socle, prime radical	C₂×C₁₁×C₆₇	C₂×C₁₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 192
transitive degree	μ_t(G) = 32428

A minimal faithful permutation representation — 1 generator on 192 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125)(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)

Conjugacy classes & character table

32428 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	Dic₁₆₂₁₄, D₃₂₄₂₈
maximal subgroup of	Dic₁₆₂₁₄, D₃₂₄₂₈
maximal normal subgroup of	Dic₁₆₂₁₄, D₃₂₄₂₈
automorphism group of	C₃₂₄₂₉
outer automorphism group of	C₃₂₄₂₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32428, 1);

C4×C67×C121 #44569812