C₂×C₇×D₁₂₄₃ #46852456

order	34804
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇×D₁₂₄₃
unambiguous semidirect product	(C₂×C₁₁)⋊(C₇×D₁₁₃)

Splittings

Fitting, socle	C₂×C₇×C₁₁×C₁₁₃ ⋊ C₂
derived, Hall	C₁₁×C₁₁₃ ⋊ C₂²×C₇
Hall	C₇×C₁₁×C₁₁₃ ⋊ C₂²
Hall	C₇×C₁₁₃ ⋊ C₂×D₁₁
Sylow	C₁₁₃ ⋊ C₂×C₇×D₁₁
Hall	C₇×C₁₁ ⋊ C₂×D₁₁₃
Sylow	C₁₁ ⋊ C₂×C₇×D₁₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₁₂₄₃
commutator, cosocle, hypocenter	C₁₁×C₁₁₃	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₁×C₁₁₃	C₂
LCS	G ▷ C₁₁×C₁₁₃
UCS	C₁ ◁ C₂×C₇

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 17402

A minimal faithful permutation representation — 2 generators on 133 points:

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

Conjugacy classes & character table

8722 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34804, 34);

C2×C7×D1243 #46852456