C₉₇×#1738 #27137696

order	12416
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₉₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉₇×#1738

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉₇	C₂×D₄
commutator	C₄	C₂×C₄²×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₉₇	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ C₂³×C₄×C₉₇ ◁ G

372 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 117
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 117 points:

(5,6,8,13,10,14,9,12)(7,15,17,11,16,20,18,19)(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);(1,2,3,4)(5,7,9,18,10,16,8,17)(6,11,13,20,14,19,12,15);(7,16)(11,19)(15,20)(17,18)

Conjugacy classes & character table

4268 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 470);