C₇×#19096903 #46828639

order	34524
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₇×#19096903

Splittings

Fitting, socle, Hall	C₃²×C₇×C₁₃₇ ⋊ C₄
derived, Hall	C₃²×C₁₃₇ ⋊ C₄×C₇
Hall	C₇×C₁₃₇ ⋊ #157
Sylow	C₁₃₇ ⋊ C₇×#157
Hall	C₃²×C₇ ⋊ #10587739
Sylow	C₃² ⋊ C₇×#10587739

Subgroups

	subgroup	G/H
center, hypercenter	C₇	#19096903
commutator, hypocenter	C₃²×C₁₃₇	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₁₃₇	C₄
prime radical	C₇×C₃⋊D₄₁₁	C₂
cosocle	C₃⋊D₄₁₁	C₂×C₇
LCS	G ▷ C₃²×C₁₃₇
UCS	C₁ ◁ C₇

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 150
transitive degree	μ_t(G) = 5754

A minimal faithful permutation representation — 2 generators on 150 points:

(1,2,3,4,5,6,7)(8,9)(10,12,13,11)(15,16,17,19)(18,21,22,28)(20,25,26,24)(23,30,31,41)(27,34,35,37)(29,38,39,33)(32,43,44,58)(36,47,48,54)(40,51,52,50)(42,55,56,46)(45,60,61,79)(49,64,65,75)(53,68,69,71)(57,72,73,67)(59,76,77,63)(62,81,82,104)(66,85,86,100)(70,89,90,96)(74,93,94,92)(78,97,98,88)(80,101,102,84)(83,106,107,128)(87,110,111,124)(91,114,115,120)(95,117,112,118)(99,119,108,116)(103,121,122,113)(105,125,126,109)(123,133,134,132)(127,136,137,131)(129,135,139,130)(138,143,144,142)(140,145,146,141)(147,149,150,148);(8,10,13)(9,12,11)(14,15,18,23,32,45,62,83,108,98,77,58,75,96,118,106,129,140,147,149,143,133,119,93,68,47,30,42,57,74,95,111,86,65,48,35,26,19,24,33,46,63,84,109,130,134,122,102,79,100,120,110,81,105,127,138,145,136,121,97,72,51,34,21,29,40,53,70,91,115,90,69,52,39,28,37,50,67,88,113,131,141,144,137,126,104,124,114,85,60,80,103,123,135,125,101,76,55,38,25,16,20,27,36,49,66,87,112,94,73,56,41,54,71,92,116,132,142,148,150,146,139,128,117,89,64,43,59,78,99,107,82,61,44,31,22,17)

Conjugacy classes & character table

2184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34524, 82);

C7×#19096903 #46828639