C₇×D₈₈₉ #27159231

order	12446
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₈₈₉
unambiguous semidirect product	C₇⋊(C₇×D₁₂₇)

Splittings

Fitting, socle, Hall	C₇²×C₁₂₇ ⋊ C₂
derived	C₇×C₁₂₇ ⋊ C₂×C₇
Sylow	C₁₂₇ ⋊ C₇×D₇
Sylow	C₇² ⋊ D₁₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₈₈₉
commutator, cosocle, hypocenter	C₇×C₁₂₇	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₁₂₇	C₂
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₇

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 1778

A minimal faithful permutation representation — 2 generators on 141 points:

(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)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129,130,131,132,133,134)(136,137)(138,139)(140,141);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(135,136,138,140,141,139,137)

Conjugacy classes & character table

3122 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×D₇×D₁₂₇, D₇×D₈₈₉, (C₇⋊D₇)⋊D₁₂₇, …
maximal normal subgroup of	C₇×D₇×D₁₂₇, D₇×D₈₈₉, (C₇⋊D₇)⋊D₁₂₇, …
maximal quotient of	C₇×Dic₈₈₉, C₂×C₇×D₈₈₉, C₃×C₇×D₈₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12446, 14);

C7×D889 #27159231