Dic₂₁×C₂₃₉ #34095507

order	20076
isoclinism	isoclinic class of D₂₁
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₃₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₂₁×C₂₃₉
unambiguous semidirect product	C₃⋊(Dic₇×C₂₃₉)
unambiguous semidirect product	C₇⋊(Dic₃×C₂₃₉)

Splittings

derived, Hall	C₃×C₇ ⋊ C₄×C₂₃₉
Hall	C₃×C₇×C₂₃₉ ⋊ C₄
Hall	C₇×C₂₃₉ ⋊ Dic₃
Hall	C₃×C₂₃₉ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₃×C₂₃₉
Sylow	C₃ ⋊ Dic₇×C₂₃₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₃₉	D₂₁
commutator, hypocenter	C₃×C₇	C₄×C₂₃₉
Frattini	C₂	D₂₁×C₂₃₉
Fitting, socle, prime radical	C₂×C₃×C₇×C₂₃₉	C₂
cosocle	C₂×C₃×C₇	C₂×C₂₃₉
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₂₃₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 253
transitive degree	μ_t(G) = 20076

A minimal faithful permutation representation — 2 generators on 253 points:

(2,3)(4,5)(6,7)(9,10)(11,14,13,12)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253);(1,2,4,6,7,5,3)(8,9,10)

Conjugacy classes & character table

5736 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20076, 9);

Dic21×C239 #34095507