D₁₅₃₂₃ #44499772

order	30646
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₃₂₃
unambiguous semidirect product	C₇⋊D₂₁₈₉
unambiguous semidirect product	C₁₁⋊D₁₃₉₃
unambiguous semidirect product	C₁₉₉⋊D₇₇

Splittings

Fitting, socle, derived, Hall	C₇×C₁₁×C₁₉₉ ⋊ C₂
Hall	C₁₁×C₁₉₉ ⋊ D₇
Hall	C₇×C₁₉₉ ⋊ D₁₁
Sylow	C₁₉₉ ⋊ D₇₇
Hall	C₇×C₁₁ ⋊ D₁₉₉
Sylow	C₁₁ ⋊ D₁₃₉₃
Sylow	C₇ ⋊ D₂₁₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₁×C₁₉₉	C₂
LCS	G ▷ C₇×C₁₁×C₁₉₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 217
transitive degree	μ_t(G) = 15323

A minimal faithful permutation representation — 2 generators on 217 points:

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

Polycyclic presentation:

2,7,11,199|b^a=b^6,c^a=c^10,d^a=d^198

Conjugacy classes & character table

7663 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₃₂₃
maximal normal subgroup of	C₂×D₁₅₃₂₃
maximal quotient of	Dic₁₅₃₂₃, C₂×D₁₅₃₂₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30646, 9);

D15323 #44499772