C₈×Dic₆₃₂ #34200004

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈×Dic₆₃₂

Splittings

Sylow

C₇₉ ⋊ C₈×Q₃₂

Subgroups

	subgroup	G/H
center	C₂×C₈	D₆₃₂
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₄×D₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	C₈×Q₃₂
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

226 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 119
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 119 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(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)(88,90,93,101)(89,94,97,111)(91,103,106,119)(92,108,104,100)(95,113,114,118)(96,115,107,110)(98,116,117,102)(99,112,109,105);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(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)(88,89,93,97)(90,98,101,117)(91,102,106,116)(92,107,104,96)(94,109,111,99)(95,112,114,105)(100,118,108,113)(103,115,119,110);(80,81,82,84,83,85,86,87)

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 81303);