D₉₇⋊#329 #27157182

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

unambiguous semidirect product

D₉₇⋊#329

Splittings

Sylow

C₉₇ ⋊ C₂×#329

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	#124×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂×D₄×D₉₇	C₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	#124	#87604
hypocenter	C₉₇	C₂×#329
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

284 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 113
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 113 points:

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

Polycyclic presentation:

2^7,97|a^2=e,d^2=g,e^2=f,b^a=bd,d^a=df,h^a=h^22,d^b=dg,e^b=efg,h^c=h^96,h^e=h^96

Conjugacy classes & character table

374 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 19956);