C₂×C₅×D₅₂₃⋊C₃ #44535938

order	31380
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₅₂₃⋊C₃
unambiguous semidirect product	D₅₂₃⋊(C₂×C₃×C₅)
unambiguous semidirect product	(C₅₂₃⋊C₃)⋊(C₂²×C₅)
unambiguous semidirect product	(C₂×D₅₂₃)⋊(C₃×C₅)
unambiguous semidirect product	(C₂×C₅₂₃⋊C₃)⋊(C₂×C₅)
unambiguous semidirect product	(C₂×C₅×D₅₂₃)⋊C₃

Splittings

Fitting, socle	C₂×C₅×C₅₂₃ ⋊ C₂×C₃
derived, Sylow	C₅₂₃ ⋊ C₂²×C₃×C₅
Hall	C₂×C₅×D₅₂₃ ⋊ C₃
Hall	C₅×C₅₂₃⋊C₃ ⋊ C₂²
Hall	C₅×C₅₂₃ ⋊ C₂²×C₃
Hall	C₂×D₅₂₃ ⋊ C₃×C₅
Hall	C₅₂₃⋊C₃ ⋊ C₂²×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₅₂₃⋊C₃
commutator, cosocle, hypocenter	C₅₂₃	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₅₂₃	C₂×C₃
LCS	G ▷ C₅₂₃
UCS	C₁ ◁ C₂×C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 530
transitive degree	μ_t(G) = 5230

A minimal faithful permutation representation — 2 generators on 530 points:

(3,4,5,6,7)(9,13,14,10,11,15)(12,21,22,16,17,31)(18,34,35,24,25,48)(19,33,29,27,23,20)(26,53,54,37,38,71)(28,51,52,40,36,45)(30,32,50,42,43,46)(39,76,77,56,57,100)(41,74,75,59,55,68)(44,49,73,61,62,69)(47,89,87,65,66,64)(58,105,106,79,80,135)(60,103,104,82,78,97)(63,72,102,84,85,98)(67,118,119,90,91,93)(70,124,116,94,95,88)(81,140,141,108,109,176)(83,138,139,111,107,132)(86,101,137,113,114,133)(92,153,154,120,121,128)(96,160,158,125,126,123)(99,165,151,129,130,117)(110,181,182,143,144,223)(112,179,180,146,142,173)(115,136,178,148,149,174)(122,194,195,155,156,169)(127,201,202,161,162,164)(131,207,199,166,167,159)(134,212,192,170,171,152)(145,228,229,184,185,276)(147,226,227,187,183,220)(150,177,225,189,190,221)(157,241,242,196,197,216)(163,248,249,203,204,211)(168,255,253,208,209,206)(172,260,246,213,214,200)(175,265,239,217,218,193)(186,281,282,231,232,335)(188,279,280,234,230,273)(191,224,278,236,237,274)(198,294,295,243,244,269)(205,301,302,250,251,264)(210,308,309,256,257,259)(215,314,306,261,262,254)(219,319,299,266,267,247)(222,324,292,270,271,240)(233,340,341,284,285,400)(235,338,339,287,283,332)(238,277,337,289,290,333)(245,353,354,296,297,328)(252,360,361,303,304,323)(258,367,368,310,311,318)(263,374,372,315,316,313)(268,379,365,320,321,307)(272,384,358,325,326,300)(275,389,351,329,330,293)(286,405,406,343,344,470)(288,403,404,346,342,397)(291,336,402,348,349,398)(298,418,419,355,356,393)(305,425,426,362,363,388)(312,432,433,369,370,383)(317,438,439,375,376,378)(322,444,436,380,381,373)(327,449,430,385,386,366)(331,454,423,390,391,359)(334,459,416,394,395,352)(345,475,476,408,409,521)(347,473,474,411,407,467)(350,401,472,413,414,468)(357,484,485,420,421,463)(364,491,410,427,428,458)(371,497,481,434,435,453)(377,500,501,440,441,448)(382,505,503,445,446,443)(387,508,498,450,451,437)(392,510,495,455,456,431)(396,512,489,460,461,424)(399,515,482,464,465,417)(412,522,523,478,477,519)(415,471,502,480,442,520)(422,524,525,486,487,517)(429,527,479,492,493,514)(447,518,483,469,506,507)(452,516,490,466,509,504)(457,513,496,462,511,499)(488,528,529,526,494,530);(1,2)(8,9,12,18,26,39,58,81,110,145,186,233,286,345,410,419,339,229,278,324,319,314,308,301,294,279,228,277,334,396,462,455,390,329,237,144,183,197,204,209,214,218,221,223,273,328,388,453,480,413,348,289,236,189,148,113,84,61,42,29,14,13,19,28,41,60,83,112,147,188,235,288,347,412,479,485,404,282,337,389,384,379,374,367,360,353,338,281,336,399,466,450,385,325,270,190,109,142,156,162,167,171,174,176,220,269,323,383,448,507,482,416,351,292,239,192,151,116,87,52,22,33,21,32,47,67,92,122,157,198,245,298,357,422,488,525,474,341,402,459,454,449,444,438,432,425,418,403,340,401,469,445,380,320,266,217,149,80,107,121,126,130,133,135,173,216,264,318,378,443,504,489,423,358,299,246,199,158,119,75,35,50,51,34,49,70,96,127,163,205,252,305,364,429,494,529,523,406,472,515,512,510,508,505,500,497,491,484,473,405,471,440,375,315,261,213,170,114,57,78,91,95,98,100,132,169,211,259,313,373,437,499,495,430,365,306,253,202,154,104,54,73,89,74,53,72,99,131,168,210,258,312,371,409,477,487,493,476,502,506,509,511,513,516,518,520,521,527,524,522,475,434,369,310,256,208,166,129,85,38,55,66,69,71,97,128,164,206,254,307,366,431,496,498,436,372,309,249,195,139,77,102,124,118,103,76,101,134,172,215,263,317,377,442,344,407,421,428,435,441,446,451,456,461,465,468,470,519,530,528,492,427,362,303,250,203,161,125,94,62,25,36,46,48,68,93,123,159,200,247,300,359,424,490,503,439,368,302,242,180,106,137,165,160,153,138,105,136,175,219,268,322,382,447,414,285,342,356,363,370,376,381,386,391,395,398,400,467,517,526,486,420,355,296,243,196,155,120,90,65,43,17,23,31,45,64,88,117,152,193,240,293,352,417,483,501,433,361,295,227,141,178,212,207,201,194,179,140,177,222,272,327,387,452,464,349,232,283,297,304,311,316,321,326,330,333,335,397,463,514,478,411,346,287,234,187,146,111,82,59,40,27,11,15,20,30,44,63,86,115,150,191,238,291,350,415,481,426,354,280,182,225,265,260,255,248,241,226,181,224,275,331,392,457,460,394,290,185,230,244,251,257,262,267,271,274,276,332,393,458,408,343,284,231,184,143,108,79,56,37,24,16,10)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31380, 17);

C2×C5×D523⋊C3 #44535938