direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.10D4, (C2×C12)⋊25D4, C24.7(C2×C6), C23.9(C3×D4), C6.92C22≀C2, (C22×D4).4C6, C22.70(C6×D4), (C22×C6).28D4, (C23×C6).6C22, C6.138(C4⋊D4), C2.C42⋊14C6, C6.68(C4.4D4), C23.81(C22×C6), (C22×C6).458C23, (C22×C12).402C22, C6.90(C22.D4), (C2×C4⋊C4)⋊5C6, (C2×C4)⋊3(C3×D4), (C6×C4⋊C4)⋊32C2, (D4×C2×C6).15C2, (C6×C22⋊C4)⋊8C2, (C2×C22⋊C4)⋊7C6, C2.7(C3×C4⋊D4), C2.6(C3×C22≀C2), (C2×C6).610(C2×D4), C2.6(C3×C4.4D4), (C22×C4).11(C2×C6), C22.37(C3×C4○D4), (C2×C6).218(C4○D4), C2.6(C3×C22.D4), (C3×C2.C42)⋊27C2, SmallGroup(192,827)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.10D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce-1 >
Subgroups: 458 in 238 conjugacy classes, 78 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C23×C6, C23.10D4, C3×C2.C42, C6×C22⋊C4, C6×C4⋊C4, D4×C2×C6, C3×C23.10D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C6×D4, C3×C4○D4, C23.10D4, C3×C22≀C2, C3×C4⋊D4, C3×C22.D4, C3×C4.4D4, C3×C23.10D4
►On 96 points
Generators in S96
(1 76 45)(2 73 46)(3 74 47)(4 75 48)(5 95 90)(6 96 91)(7 93 92)(8 94 89)(9 17 16)(10 18 13)(11 19 14)(12 20 15)(21 51 56)(22 52 53)(23 49 54)(24 50 55)(25 68 32)(26 65 29)(27 66 30)(28 67 31)(33 39 41)(34 40 42)(35 37 43)(36 38 44)(57 62 70)(58 63 71)(59 64 72)(60 61 69)(77 88 83)(78 85 84)(79 86 81)(80 87 82)
(1 84)(2 67)(3 82)(4 65)(5 20)(6 49)(7 18)(8 51)(9 62)(10 92)(11 64)(12 90)(13 93)(14 59)(15 95)(16 57)(17 70)(19 72)(21 89)(22 61)(23 91)(24 63)(25 42)(26 48)(27 44)(28 46)(29 75)(30 38)(31 73)(32 40)(33 81)(34 68)(35 83)(36 66)(37 77)(39 79)(41 86)(43 88)(45 85)(47 87)(50 71)(52 69)(53 60)(54 96)(55 58)(56 94)(74 80)(76 78)
(1 22)(2 23)(3 24)(4 21)(5 30)(6 31)(7 32)(8 29)(9 33)(10 34)(11 35)(12 36)(13 42)(14 43)(15 44)(16 41)(17 39)(18 40)(19 37)(20 38)(25 93)(26 94)(27 95)(28 96)(45 53)(46 54)(47 55)(48 56)(49 73)(50 74)(51 75)(52 76)(57 86)(58 87)(59 88)(60 85)(61 84)(62 81)(63 82)(64 83)(65 89)(66 90)(67 91)(68 92)(69 78)(70 79)(71 80)(72 77)
(1 12)(2 9)(3 10)(4 11)(5 78)(6 79)(7 80)(8 77)(13 47)(14 48)(15 45)(16 46)(17 73)(18 74)(19 75)(20 76)(21 35)(22 36)(23 33)(24 34)(25 58)(26 59)(27 60)(28 57)(29 72)(30 69)(31 70)(32 71)(37 51)(38 52)(39 49)(40 50)(41 54)(42 55)(43 56)(44 53)(61 66)(62 67)(63 68)(64 65)(81 91)(82 92)(83 89)(84 90)(85 95)(86 96)(87 93)(88 94)
(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 12)(2 35)(3 10)(4 33)(5 69)(6 77)(7 71)(8 79)(9 21)(11 23)(13 47)(14 54)(15 45)(16 56)(17 51)(18 74)(19 49)(20 76)(22 36)(24 34)(25 87)(26 57)(27 85)(28 59)(29 70)(30 78)(31 72)(32 80)(37 73)(38 52)(39 75)(40 50)(41 48)(42 55)(43 46)(44 53)(58 93)(60 95)(61 90)(62 65)(63 92)(64 67)(66 84)(68 82)(81 89)(83 91)(86 94)(88 96)
G:=sub<Sym(96)| (1,76,45)(2,73,46)(3,74,47)(4,75,48)(5,95,90)(6,96,91)(7,93,92)(8,94,89)(9,17,16)(10,18,13)(11,19,14)(12,20,15)(21,51,56)(22,52,53)(23,49,54)(24,50,55)(25,68,32)(26,65,29)(27,66,30)(28,67,31)(33,39,41)(34,40,42)(35,37,43)(36,38,44)(57,62,70)(58,63,71)(59,64,72)(60,61,69)(77,88,83)(78,85,84)(79,86,81)(80,87,82), (1,84)(2,67)(3,82)(4,65)(5,20)(6,49)(7,18)(8,51)(9,62)(10,92)(11,64)(12,90)(13,93)(14,59)(15,95)(16,57)(17,70)(19,72)(21,89)(22,61)(23,91)(24,63)(25,42)(26,48)(27,44)(28,46)(29,75)(30,38)(31,73)(32,40)(33,81)(34,68)(35,83)(36,66)(37,77)(39,79)(41,86)(43,88)(45,85)(47,87)(50,71)(52,69)(53,60)(54,96)(55,58)(56,94)(74,80)(76,78), (1,22)(2,23)(3,24)(4,21)(5,30)(6,31)(7,32)(8,29)(9,33)(10,34)(11,35)(12,36)(13,42)(14,43)(15,44)(16,41)(17,39)(18,40)(19,37)(20,38)(25,93)(26,94)(27,95)(28,96)(45,53)(46,54)(47,55)(48,56)(49,73)(50,74)(51,75)(52,76)(57,86)(58,87)(59,88)(60,85)(61,84)(62,81)(63,82)(64,83)(65,89)(66,90)(67,91)(68,92)(69,78)(70,79)(71,80)(72,77), (1,12)(2,9)(3,10)(4,11)(5,78)(6,79)(7,80)(8,77)(13,47)(14,48)(15,45)(16,46)(17,73)(18,74)(19,75)(20,76)(21,35)(22,36)(23,33)(24,34)(25,58)(26,59)(27,60)(28,57)(29,72)(30,69)(31,70)(32,71)(37,51)(38,52)(39,49)(40,50)(41,54)(42,55)(43,56)(44,53)(61,66)(62,67)(63,68)(64,65)(81,91)(82,92)(83,89)(84,90)(85,95)(86,96)(87,93)(88,94), (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), (1,12)(2,35)(3,10)(4,33)(5,69)(6,77)(7,71)(8,79)(9,21)(11,23)(13,47)(14,54)(15,45)(16,56)(17,51)(18,74)(19,49)(20,76)(22,36)(24,34)(25,87)(26,57)(27,85)(28,59)(29,70)(30,78)(31,72)(32,80)(37,73)(38,52)(39,75)(40,50)(41,48)(42,55)(43,46)(44,53)(58,93)(60,95)(61,90)(62,65)(63,92)(64,67)(66,84)(68,82)(81,89)(83,91)(86,94)(88,96)>;
G:=Group( (1,76,45)(2,73,46)(3,74,47)(4,75,48)(5,95,90)(6,96,91)(7,93,92)(8,94,89)(9,17,16)(10,18,13)(11,19,14)(12,20,15)(21,51,56)(22,52,53)(23,49,54)(24,50,55)(25,68,32)(26,65,29)(27,66,30)(28,67,31)(33,39,41)(34,40,42)(35,37,43)(36,38,44)(57,62,70)(58,63,71)(59,64,72)(60,61,69)(77,88,83)(78,85,84)(79,86,81)(80,87,82), (1,84)(2,67)(3,82)(4,65)(5,20)(6,49)(7,18)(8,51)(9,62)(10,92)(11,64)(12,90)(13,93)(14,59)(15,95)(16,57)(17,70)(19,72)(21,89)(22,61)(23,91)(24,63)(25,42)(26,48)(27,44)(28,46)(29,75)(30,38)(31,73)(32,40)(33,81)(34,68)(35,83)(36,66)(37,77)(39,79)(41,86)(43,88)(45,85)(47,87)(50,71)(52,69)(53,60)(54,96)(55,58)(56,94)(74,80)(76,78), (1,22)(2,23)(3,24)(4,21)(5,30)(6,31)(7,32)(8,29)(9,33)(10,34)(11,35)(12,36)(13,42)(14,43)(15,44)(16,41)(17,39)(18,40)(19,37)(20,38)(25,93)(26,94)(27,95)(28,96)(45,53)(46,54)(47,55)(48,56)(49,73)(50,74)(51,75)(52,76)(57,86)(58,87)(59,88)(60,85)(61,84)(62,81)(63,82)(64,83)(65,89)(66,90)(67,91)(68,92)(69,78)(70,79)(71,80)(72,77), (1,12)(2,9)(3,10)(4,11)(5,78)(6,79)(7,80)(8,77)(13,47)(14,48)(15,45)(16,46)(17,73)(18,74)(19,75)(20,76)(21,35)(22,36)(23,33)(24,34)(25,58)(26,59)(27,60)(28,57)(29,72)(30,69)(31,70)(32,71)(37,51)(38,52)(39,49)(40,50)(41,54)(42,55)(43,56)(44,53)(61,66)(62,67)(63,68)(64,65)(81,91)(82,92)(83,89)(84,90)(85,95)(86,96)(87,93)(88,94), (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), (1,12)(2,35)(3,10)(4,33)(5,69)(6,77)(7,71)(8,79)(9,21)(11,23)(13,47)(14,54)(15,45)(16,56)(17,51)(18,74)(19,49)(20,76)(22,36)(24,34)(25,87)(26,57)(27,85)(28,59)(29,70)(30,78)(31,72)(32,80)(37,73)(38,52)(39,75)(40,50)(41,48)(42,55)(43,46)(44,53)(58,93)(60,95)(61,90)(62,65)(63,92)(64,67)(66,84)(68,82)(81,89)(83,91)(86,94)(88,96) );
G=PermutationGroup([[(1,76,45),(2,73,46),(3,74,47),(4,75,48),(5,95,90),(6,96,91),(7,93,92),(8,94,89),(9,17,16),(10,18,13),(11,19,14),(12,20,15),(21,51,56),(22,52,53),(23,49,54),(24,50,55),(25,68,32),(26,65,29),(27,66,30),(28,67,31),(33,39,41),(34,40,42),(35,37,43),(36,38,44),(57,62,70),(58,63,71),(59,64,72),(60,61,69),(77,88,83),(78,85,84),(79,86,81),(80,87,82)], [(1,84),(2,67),(3,82),(4,65),(5,20),(6,49),(7,18),(8,51),(9,62),(10,92),(11,64),(12,90),(13,93),(14,59),(15,95),(16,57),(17,70),(19,72),(21,89),(22,61),(23,91),(24,63),(25,42),(26,48),(27,44),(28,46),(29,75),(30,38),(31,73),(32,40),(33,81),(34,68),(35,83),(36,66),(37,77),(39,79),(41,86),(43,88),(45,85),(47,87),(50,71),(52,69),(53,60),(54,96),(55,58),(56,94),(74,80),(76,78)], [(1,22),(2,23),(3,24),(4,21),(5,30),(6,31),(7,32),(8,29),(9,33),(10,34),(11,35),(12,36),(13,42),(14,43),(15,44),(16,41),(17,39),(18,40),(19,37),(20,38),(25,93),(26,94),(27,95),(28,96),(45,53),(46,54),(47,55),(48,56),(49,73),(50,74),(51,75),(52,76),(57,86),(58,87),(59,88),(60,85),(61,84),(62,81),(63,82),(64,83),(65,89),(66,90),(67,91),(68,92),(69,78),(70,79),(71,80),(72,77)], [(1,12),(2,9),(3,10),(4,11),(5,78),(6,79),(7,80),(8,77),(13,47),(14,48),(15,45),(16,46),(17,73),(18,74),(19,75),(20,76),(21,35),(22,36),(23,33),(24,34),(25,58),(26,59),(27,60),(28,57),(29,72),(30,69),(31,70),(32,71),(37,51),(38,52),(39,49),(40,50),(41,54),(42,55),(43,56),(44,53),(61,66),(62,67),(63,68),(64,65),(81,91),(82,92),(83,89),(84,90),(85,95),(86,96),(87,93),(88,94)], [(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)], [(1,12),(2,35),(3,10),(4,33),(5,69),(6,77),(7,71),(8,79),(9,21),(11,23),(13,47),(14,54),(15,45),(16,56),(17,51),(18,74),(19,49),(20,76),(22,36),(24,34),(25,87),(26,57),(27,85),(28,59),(29,70),(30,78),(31,72),(32,80),(37,73),(38,52),(39,75),(40,50),(41,48),(42,55),(43,46),(44,53),(58,93),(60,95),(61,90),(62,65),(63,92),(64,67),(66,84),(68,82),(81,89),(83,91),(86,94),(88,96)]])
66 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4J | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 |
| kernel | C3×C23.10D4 | C3×C2.C42 | C6×C22⋊C4 | C6×C4⋊C4 | D4×C2×C6 | C23.10D4 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C22×D4 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 4 | 6 | 8 | 8 | 12 |
Matrix representation of C3×C23.10D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 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 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;
C3×C23.10D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{10}D_4 % in TeX
G:=Group("C3xC2^3.10D4"); // GroupNames label
G:=SmallGroup(192,827);
// by ID
G=gap.SmallGroup(192,827);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,1059,142]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,b*d=d*b,e*b*e^-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=c*e^-1>;
// generators/relations