C₂×C₇×D₁₀₈₉ #44478809

order	30492
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇×D₁₀₈₉
unambiguous semidirect product	(C₂×C₉)⋊(C₇×D₁₂₁)
unambiguous semidirect product	(C₂×C₁₂₁)⋊(C₇×D₉)

Splittings

Fitting	C₂×C₇×C₉×C₁₂₁ ⋊ C₂
derived, Hall	C₉×C₁₂₁ ⋊ C₂²×C₇
Hall	C₇×C₉×C₁₂₁ ⋊ C₂²
Hall	C₇×C₁₂₁ ⋊ C₂×D₉
Sylow	C₁₂₁ ⋊ C₂×C₇×D₉
Hall	C₇×C₉ ⋊ C₂×D₁₂₁
Sylow	C₉ ⋊ C₂×C₇×D₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₁₀₈₉
commutator, cosocle, hypocenter	C₉×C₁₂₁	C₂²×C₇
Frattini	C₃×C₁₁	C₂×C₇×D₃₃
Fitting	C₂×C₇×C₉×C₁₂₁	C₂
socle	C₂×C₃×C₇×C₁₁	D₃₃
LCS	G ▷ C₉×C₁₂₁
UCS	C₁ ◁ C₂×C₇

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 139
transitive degree	μ_t(G) = 15246

A minimal faithful permutation representation — 2 generators on 139 points:

(3,4,5,6,7,8,9)(11,13)(12,16)(14,15)(17,18)(20,22)(21,25)(23,28)(24,31)(26,34)(27,33)(29,38)(30,41)(32,43)(35,47)(36,46)(37,45)(39,52)(40,55)(42,57)(44,59)(48,64)(49,63)(50,62)(51,61)(53,70)(54,73)(56,75)(58,77)(60,79)(65,85)(66,84)(67,83)(68,82)(69,81)(71,72)(74,92)(76,94)(78,96)(80,98)(86,99)(87,103)(88,102)(89,101)(90,100)(91,108)(93,110)(95,112)(97,114)(104,115)(105,118)(106,117)(107,116)(109,122)(111,124)(113,126)(119,127)(120,129)(121,128)(123,132)(125,134)(130,135)(131,136)(133,138)(137,139);(1,2)(10,18,13,16,15,14,12,11,17)(19,63,97,56,81,136,121,52,94,104,36,25,84,78,40,101,139,107,73,112,87,27,34,99,58,29,117,133,90,92,126,67,22,47,80,42,61,129,123,70,110,119,50,31,64,60,30,82,135,109,71,124,105,37,43,85,44,23,102,125,91,108,134,88,28,59,65,32,45,118,111,72,122,130,68,41,79,48,24,62,127,93,53,132,120,51,57,98,35,20,83,113,74,100,138,106,38,77,86,26,33,103,95,54,116,137,89,55,96,66,21,46,115,76,39,128,131,69,75,114,49)

Conjugacy classes & character table

7644 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 32);

C2×C7×D1089 #44478809