C₂×#20261828 #27158890

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#20261828

Splittings

derived, Sylow

C₉₇ ⋊ C₂×C₄×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	#11715906
commutator, hypocenter	C₉₇	C₂×C₄×C₁₆
Frattini	C₄	C₂²×#11715906
Fitting	C₂×C₈×C₉₇	C₈
socle	C₂²×C₉₇	C₄×C₈
prime radical	C₂²×D₉₇	C₂×C₈
cosocle	#13095519	C₂³
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₈

216 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 115
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21664);