C₉₇×#7074 #37614040

order	24832
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉₇×#7074

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉₇	#102
commutator	C₂×C₈	C₂×C₈×C₉₇
Frattini, cosocle	C₈²	C₂²×C₉₇
socle	C₂²×C₉₇	D₄⋊C₈
prime radical	C₂×D₈×C₉₇	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ C₄×C₈×C₉₇ ◁ C₈²×C₉₇ ◁ G

148 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 129
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 2 generators on 129 points:

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

Conjugacy classes & character table

5626 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24832, 62);