C₂×C₇⋊(D₂₄₁⋊C₅) #46798096

order	33740
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇⋊(D₂₄₁⋊C₅)
unambiguous semidirect product	(C₂×C₇)⋊(D₂₄₁⋊C₅)
unambiguous semidirect product	(C₂×C₂₄₁⋊C₅)⋊D₇
unambiguous semidirect product	(C₂×D₁₆₈₇)⋊C₅

Splittings

Fitting, socle	C₂×C₇×C₂₄₁ ⋊ C₂×C₅
derived, Hall	C₇×C₂₄₁ ⋊ C₂²×C₅
Hall	C₇×C₂₄₁⋊C₅ ⋊ C₂²
Hall	C₂×D₁₆₈₇ ⋊ C₅
Hall	C₂₄₁⋊C₅ ⋊ C₂×D₇
Sylow	C₂₄₁ ⋊ C₂×C₅×D₇
Sylow	C₇ ⋊ C₂×D₂₄₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₇⋊(D₂₄₁⋊C₅)
commutator, cosocle, hypocenter	C₇×C₂₄₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₂₄₁	C₂×C₅
LCS	G ▷ C₇×C₂₄₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 250
transitive degree	μ_t(G) = 3374

A minimal faithful permutation representation — 2 generators on 250 points:

(4,5)(6,7)(8,9)(11,15,16,34,38,12,13,21,23,62)(14,28,29,87,92,18,19,49,51,145)(17,43,44,45,68,25,26,76,78,80)(20,56,57,168,134,31,32,102,104,200)(22,59,60,170,177,36,33,109,110,115)(24,72,73,69,70,40,41,74,39,79)(27,82,83,183,153,46,47,136,137,223)(30,97,98,230,126,53,54,160,161,197)(35,119,120,63,185,64,65,189,116,117)(37,75,103,93,224,42,67,129,154,155)(48,142,143,233,199,85,86,162,190,128)(50,150,151,240,125,89,90,221,222,186)(52,157,158,108,127,94,95,227,71,176)(55,165,166,178,179,100,101,138,111,229)(58,174,171,172,198,106,107,105,234,207)(61,181,182,81,152,112,113,225,91,135)(66,194,195,146,246,122,123,241,212,213)(77,203,99,235,114,131,132,163,164,184)(84,208,191,192,130,139,140,121,232,201)(88,215,216,118,124,147,148,237,187,188)(96,205,193,220,226,133,159,175,156,206)(141,238,245,167,239,169,209,204,180,236)(144,247,242,228,202,210,211,217,250,244)(149,248,196,214,249,218,219,173,243,231);(1,2)(3,4,6,8,9,7,5)(10,11,14,20,33,61,114,185,152,222,233,189,208,219,249,98,162,234,250,95,57,105,173,194,54,99,137,181,47,84,141,129,32,58,108,69,34,63,118,127,132,204,158,229,223,145,155,225,239,244,201,44,79,134,199,41,74,100,53,28,50,91,78,43,77,133,94,49,88,149,220,154,87,146,214,211,195,242,215,209,192,230,221,150,188,122,64,16,24,42,36,13,17,27,48,67,125,177,38,68,126,198,166,136,206,246,164,110,168,237,159,107,175,191,119,101,167,178,109,86,144,210,142,60,111,180,165,65,121,193,174,205,216,104,170,235,213,226,83,138,207,197,80,62,115,186,75,85,46,25,15,22,37,40,21,35,66,124,90,151,161,232,238,148,217,241,247,243,212,51,93,156,218,147,29,52,96,131,26,45,81,89,19,30,55,73,72,128,200,70,76,130,202,236,182,224,92,153,179,227,245,203,176,187,116,23,39,71,106,56,103,169,139,82,113,183,163,97,123,196,171,102,157,228,172,143,160,231,248,140,120,190,240,135,117,184,112,59,31,18,12)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33740, 18);

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