(C₂×#13095525)⋊C₄ #27159002

order	12416
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

unambiguous semidirect product

(C₂×#13095525)⋊C₄

Splittings

Fitting	C₂×C₄×C₉₇ ⋊ C₁₆
Sylow	C₉₇ ⋊ #1374

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×#13095525
commutator	C₂×C₉₇	C₄×C₁₆
Fitting	C₂×C₄×C₉₇	C₁₆
socle	C₂²×C₉₇	C₂×C₁₆
prime radical	C₂²×D₉₇	C₂×C₈
cosocle	C₂²×#11715906	C₂²
hypercenter	C₂×C₄	#13095525
hypocenter	C₉₇	#1374
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

160 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 105
transitive degree	μ_t(G) = 776

A minimal faithful permutation representation — 2 generators on 105 points:

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

Polycyclic presentation:

2^7,97|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bc,h^a=h^8,h^d=h^64,h^f=h^22,h^g=h^96

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21776);