C₂×D₃₇×C₈₉ #27190604

order	13172
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃₇×C₈₉
unambiguous semidirect product	C₃₇⋊(C₂²×C₈₉)
unambiguous semidirect product	(C₂×C₃₇)⋊(C₂×C₈₉)

Splittings

Fitting, socle	C₂×C₃₇×C₈₉ ⋊ C₂
derived, Sylow	C₃₇ ⋊ C₂²×C₈₉
Hall	C₃₇×C₈₉ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈₉	D₃₇
commutator, cosocle, hypocenter	C₃₇	C₂²×C₈₉
Frattini	C₁	G
Fitting, socle	C₂×C₃₇×C₈₉	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×C₈₉

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 6586

A minimal faithful permutation representation — 2 generators on 128 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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(127,128)

Conjugacy classes & character table

3560 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13172, 14);

C2×D37×C89 #27190604