C₂×#15692423⋊C₂ #27188665

order	12996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×#15692423⋊C₂
unambiguous semidirect product	#15692423⋊C₂²
unambiguous semidirect product	(C₂×#15692423)⋊C₂

Splittings

derived, Sylow	C₃₆₁ ⋊ C₂²×C₉
Hall	#15692423 ⋊ C₂²
Hall	C₂×D₃₆₁ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃₆₁⋊C₃
commutator, hypocenter	C₃₆₁	C₂²×C₉
Frattini	C₃×C₁₉	C₂×D₁₉⋊C₃
Fitting	C₂×C₃×C₃₆₁	C₂×C₃
socle	C₂×C₃×C₁₉	D₁₉⋊C₃
prime radical	C₂×C₃×D₃₆₁	C₃
cosocle	C₃×C₃₆₁	C₂²×C₃
LCS	G ▷ C₃₆₁
UCS	C₁ ◁ C₂×C₃

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 372
transitive degree	μ_t(G) = 6498

A minimal faithful permutation representation — 2 generators on 372 points:

(2,11,13,4,5,14)(3,23,25,8,9,27)(6,41,43,16,17,100)(7,53,54,20,21,125)(10,70,30,28,29,74)(12,62,81,24,33,96)(15,94,64,38,39,144)(18,110,45,46,47,108)(19,119,120,50,51,220)(22,133,134,57,58,243)(26,83,35,65,66,86)(31,105,88,76,44,36)(32,129,130,79,55,68)(34,153,172,71,84,267)(37,60,128,91,92,56)(40,189,59,97,98,137)(42,142,117,63,103,49)(48,181,159,114,115,95)(52,226,227,122,123,334)(61,75,151,140,73,69)(67,202,106,146,147,206)(72,192,87,155,99,177)(77,195,113,160,112,194)(78,224,225,163,121,231)(80,247,85,165,135,158)(82,116,178,167,168,176)(89,101,107,179,180,196)(90,126,222,182,183,161)(93,152,166,185,186,251)(102,223,262,198,199,149)(104,268,313,154,203,350)(109,259,124,209,148,229)(111,230,275,143,212,162)(118,258,323,217,218,315)(127,246,264,234,235,150)(131,170,205,237,238,283)(132,187,138,240,241,252)(136,232,139,249,250,253)(141,287,265,245,191,257)(145,193,291,254,174,188)(156,322,207,270,210,319)(157,171,236,175,242,285)(164,337,338,277,228,342)(169,332,214,280,219,318)(173,356,294,288,289,293)(184,244,335,297,298,343)(190,260,355,273,300,286)(197,215,346,304,256,328)(200,269,204,307,213,272)(201,274,295,309,310,292)(208,299,278,320,239,311)(211,255,349,324,325,261)(216,341,358,329,330,290)(221,336,326,301,327,263)(233,351,352,321,281,339)(248,302,317,345,314,360)(266,308,353,306,282,303)(271,312,331,316,276,347)(279,357,359,344,348,296)(284,333,361,305,354,340)(362,366,370,364,369,365,367,363,368);(1,24,154,336,335,200,197,56,55,330,237,54,240,284,282,281,213,42,4,8,71,272,227,317,102,101,21,285,299,225,250,171,170,169,228,111,16,20,28,158,271,207,205,41,183,234,189,346,352,188,83,82,123,215,46,50,57,74,157,156,106,105,108,321,153,287,286,296,96,33,344,255,114,117,122,137,73,72,214,43,196,209,288,193,172,184,181,39,245,244,142,168,217,229,252,29,139,113,112,320,322,358,99,93,90,89,241,133,309,147,329,332,212,186,61,60,47,353,356,315,314,40,37,36,292,247,249,278,276,274,220,306,145,23,64,324,350,348,203,273,15,14,176,345,246,166,164,162,161,199,258,65,25,265,263,261,260,221,159,5,86,293,361,134,80,78,77,121,347,146,130,151,150,149,148,333,119,76,79,177,342,338,87,32,31,51,354,259,262,264,69,68,67,331,224,160,163,165,243,340,294,35,11,95,301,300,349,326,257,27,26,218,223,222,275,277,251,235,248,178,13,38,190,268,357,313,211,144,9,254,266,120,310,312,311,136,135,295,88,91,97,302,323,289,303,110,92,140,152,230,219,216,202,201,58,187,179,182,185,192,290,210,208,195,194,253,70,138,124,118,116,103,298,141,94,115,297,267,174,173,109,107,100,318,155,75,59,52,49,48,325,279,62,81,359,355,191,84,233,45,44,206,270,175,30,22,19,18,256,226,167,66,291,339,328,98,127,126,17,283,319,316,85,10,7,6,143,337,280,238,242,232,231,239,236,53,180,198,360,334,204,34,3,2,63,269,351,308,305,132,125,131,341,129,128,304,307,343,327,104,12)(371,372)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 8);

C2×#15692423⋊C2 #27188665