C₉₇×#7079 #37614045

order	24832
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉₇×#7079

Subgroups

	subgroup	G/H
center, socle	C₂²×C₉₇	#327
commutator	C₂²×C₈	C₂×C₄×C₉₇
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₉₇
prime radical	C₂⁴×C₉₇	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ C₂×C₂²⋊C₈×C₉₇ ◁ G

218 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 6208

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

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

Conjugacy classes & character table

3298 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24832, 67);