direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.11D4, C24.9(C2×C6), (C2×C12).310D4, C22.73(C6×D4), (C22×C6).29D4, C23.10(C3×D4), C2.C42⋊8C6, (C23×C6).8C22, C6.140(C4⋊D4), C6.69(C4.4D4), C23.84(C22×C6), C6.35(C42⋊2C2), (C22×C12).35C22, (C22×C6).461C23, C6.91(C22.D4), (C2×C4⋊C4)⋊7C6, (C6×C4⋊C4)⋊34C2, (C2×C4).17(C3×D4), C2.9(C3×C4⋊D4), (C2×C6).613(C2×D4), (C2×C22⋊C4).7C6, C2.7(C3×C4.4D4), (C6×C22⋊C4).29C2, (C22×C4).13(C2×C6), C2.5(C3×C42⋊2C2), C22.40(C3×C4○D4), (C2×C6).221(C4○D4), (C3×C2.C42)⋊7C2, C2.7(C3×C22.D4), SmallGroup(192,830)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.11D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 314 in 170 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C23×C6, C23.11D4, C3×C2.C42, C3×C2.C42, C6×C22⋊C4, C6×C22⋊C4, C6×C4⋊C4, C3×C23.11D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C6×D4, C3×C4○D4, C23.11D4, C3×C4⋊D4, C3×C22.D4, C3×C4.4D4, C3×C42⋊2C2, C3×C23.11D4
►On 96 points
Generators in S96
(1 15 11)(2 16 12)(3 13 9)(4 14 10)(5 69 65)(6 70 66)(7 71 67)(8 72 68)(17 25 21)(18 26 22)(19 27 23)(20 28 24)(29 37 33)(30 38 34)(31 39 35)(32 40 36)(41 52 45)(42 49 46)(43 50 47)(44 51 48)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(2 56)(4 54)(5 52)(6 94)(7 50)(8 96)(10 58)(12 60)(14 62)(16 64)(17 85)(18 42)(19 87)(20 44)(21 89)(22 46)(23 91)(24 48)(25 93)(26 49)(27 95)(28 51)(30 74)(32 76)(34 78)(36 80)(38 82)(40 84)(41 65)(43 67)(45 69)(47 71)(66 86)(68 88)(70 90)(72 92)
(1 55)(2 56)(3 53)(4 54)(5 25)(6 26)(7 27)(8 28)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 94)(50 95)(51 96)(52 93)
(1 75)(2 76)(3 73)(4 74)(5 52)(6 49)(7 50)(8 51)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)
(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)
(1 66 55 18)(2 65 56 17)(3 68 53 20)(4 67 54 19)(5 64 25 16)(6 63 26 15)(7 62 27 14)(8 61 28 13)(9 72 57 24)(10 71 58 23)(11 70 59 22)(12 69 60 21)(29 88 73 44)(30 87 74 43)(31 86 75 42)(32 85 76 41)(33 92 77 48)(34 91 78 47)(35 90 79 46)(36 89 80 45)(37 96 81 51)(38 95 82 50)(39 94 83 49)(40 93 84 52)
G:=sub<Sym(96)| (1,15,11)(2,16,12)(3,13,9)(4,14,10)(5,69,65)(6,70,66)(7,71,67)(8,72,68)(17,25,21)(18,26,22)(19,27,23)(20,28,24)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,52,45)(42,49,46)(43,50,47)(44,51,48)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (2,56)(4,54)(5,52)(6,94)(7,50)(8,96)(10,58)(12,60)(14,62)(16,64)(17,85)(18,42)(19,87)(20,44)(21,89)(22,46)(23,91)(24,48)(25,93)(26,49)(27,95)(28,51)(30,74)(32,76)(34,78)(36,80)(38,82)(40,84)(41,65)(43,67)(45,69)(47,71)(66,86)(68,88)(70,90)(72,92), (1,55)(2,56)(3,53)(4,54)(5,25)(6,26)(7,27)(8,28)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92)(49,94)(50,95)(51,96)(52,93), (1,75)(2,76)(3,73)(4,74)(5,52)(6,49)(7,50)(8,51)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (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), (1,66,55,18)(2,65,56,17)(3,68,53,20)(4,67,54,19)(5,64,25,16)(6,63,26,15)(7,62,27,14)(8,61,28,13)(9,72,57,24)(10,71,58,23)(11,70,59,22)(12,69,60,21)(29,88,73,44)(30,87,74,43)(31,86,75,42)(32,85,76,41)(33,92,77,48)(34,91,78,47)(35,90,79,46)(36,89,80,45)(37,96,81,51)(38,95,82,50)(39,94,83,49)(40,93,84,52)>;
G:=Group( (1,15,11)(2,16,12)(3,13,9)(4,14,10)(5,69,65)(6,70,66)(7,71,67)(8,72,68)(17,25,21)(18,26,22)(19,27,23)(20,28,24)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,52,45)(42,49,46)(43,50,47)(44,51,48)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (2,56)(4,54)(5,52)(6,94)(7,50)(8,96)(10,58)(12,60)(14,62)(16,64)(17,85)(18,42)(19,87)(20,44)(21,89)(22,46)(23,91)(24,48)(25,93)(26,49)(27,95)(28,51)(30,74)(32,76)(34,78)(36,80)(38,82)(40,84)(41,65)(43,67)(45,69)(47,71)(66,86)(68,88)(70,90)(72,92), (1,55)(2,56)(3,53)(4,54)(5,25)(6,26)(7,27)(8,28)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92)(49,94)(50,95)(51,96)(52,93), (1,75)(2,76)(3,73)(4,74)(5,52)(6,49)(7,50)(8,51)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (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), (1,66,55,18)(2,65,56,17)(3,68,53,20)(4,67,54,19)(5,64,25,16)(6,63,26,15)(7,62,27,14)(8,61,28,13)(9,72,57,24)(10,71,58,23)(11,70,59,22)(12,69,60,21)(29,88,73,44)(30,87,74,43)(31,86,75,42)(32,85,76,41)(33,92,77,48)(34,91,78,47)(35,90,79,46)(36,89,80,45)(37,96,81,51)(38,95,82,50)(39,94,83,49)(40,93,84,52) );
G=PermutationGroup([[(1,15,11),(2,16,12),(3,13,9),(4,14,10),(5,69,65),(6,70,66),(7,71,67),(8,72,68),(17,25,21),(18,26,22),(19,27,23),(20,28,24),(29,37,33),(30,38,34),(31,39,35),(32,40,36),(41,52,45),(42,49,46),(43,50,47),(44,51,48),(53,61,57),(54,62,58),(55,63,59),(56,64,60),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(2,56),(4,54),(5,52),(6,94),(7,50),(8,96),(10,58),(12,60),(14,62),(16,64),(17,85),(18,42),(19,87),(20,44),(21,89),(22,46),(23,91),(24,48),(25,93),(26,49),(27,95),(28,51),(30,74),(32,76),(34,78),(36,80),(38,82),(40,84),(41,65),(43,67),(45,69),(47,71),(66,86),(68,88),(70,90),(72,92)], [(1,55),(2,56),(3,53),(4,54),(5,25),(6,26),(7,27),(8,28),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88),(45,89),(46,90),(47,91),(48,92),(49,94),(50,95),(51,96),(52,93)], [(1,75),(2,76),(3,73),(4,74),(5,52),(6,49),(7,50),(8,51),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72)], [(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)], [(1,66,55,18),(2,65,56,17),(3,68,53,20),(4,67,54,19),(5,64,25,16),(6,63,26,15),(7,62,27,14),(8,61,28,13),(9,72,57,24),(10,71,58,23),(11,70,59,22),(12,69,60,21),(29,88,73,44),(30,87,74,43),(31,86,75,42),(32,85,76,41),(33,92,77,48),(34,91,78,47),(35,90,79,46),(36,89,80,45),(37,96,81,51),(38,95,82,50),(39,94,83,49),(40,93,84,52)]])
66 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6N | 6O | 6P | 6Q | 6R | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 |
| kernel | C3×C23.11D4 | C3×C2.C42 | C6×C22⋊C4 | C6×C4⋊C4 | C23.11D4 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 2 | 2 | 10 | 4 | 4 | 20 |
Matrix representation of C3×C23.11D4 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 8 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,4,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,4,1,0,0,0,0,11,9,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,6,8,0,0,0,0,10,7,0,0,0,0,0,0,8,0,0,0,0,0,0,5] >;
C3×C23.11D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{11}D_4 % in TeX
G:=Group("C3xC2^3.11D4"); // GroupNames label
G:=SmallGroup(192,830);
// by ID
G=gap.SmallGroup(192,830);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,1094,1059,142]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations