C₄×#19098348 #34203320

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×#19098348

Splittings

Fitting	C₄×M₃₂×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ C₄×#350

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	C₄×#350
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 115 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)(84,86)(85,91)(87,100)(88,102)(89,97)(90,99)(92,108)(93,110)(94,96)(95,107)(98,104)(101,114)(103,105)(106,112)(109,113)(111,115);(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)(84,85,88,93,89,94,103,111,90,95,104,112,87,92,101,109)(86,96,98,113,97,107,114,110,99,108,102,115,100,91,105,106);(80,81,82,83)

Conjugacy classes & character table

3232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 84619);

C4×#19098348 #34203320