C₂×#23761489 #34203138

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#23761489

Splittings

Sylow

C₇₉ ⋊ C₂×#1379

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#405×C₇₉	C₂
socle	C₂³×C₇₉	C₂²⋊C₈
prime radical	C₂⁴×C₇₉	M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	C₂×#1379
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂×#405

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 117 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(82,83,84,85)(86,88)(87,93)(89,102)(90,104)(91,99)(92,101)(94,110)(95,112)(96,98)(97,109)(100,106)(103,116)(105,107)(108,114)(111,115)(113,117);(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)(86,87,90,95,91,96,105,113,92,97,106,114,89,94,103,111)(88,98,100,115,99,109,116,112,101,110,104,117,102,93,107,108);(80,81)

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, 84437);

C2×#23761489 #34203138