direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.12D6, C24.48D6, (C2×D4).229D6, (C2×C12).209D4, C12.251(C2×D4), C6⋊3(C4.4D4), (C2×C6).292C24, (C22×D4).12S3, C6.140(C22×D4), (C22×C4).394D6, (C2×C12).540C23, (C2×Dic6)⋊67C22, (C22×Dic6)⋊20C2, (C4×Dic3)⋊67C22, (C6×D4).269C22, (C23×C6).74C22, C6.D4⋊58C22, (C22×C6).228C23, C22.306(S3×C23), C23.144(C22×S3), C22.78(D4⋊2S3), (C22×C12).273C22, (C2×Dic3).282C23, (C22×Dic3).231C22, (D4×C2×C6).8C2, C3⋊4(C2×C4.4D4), (C2×C4×Dic3)⋊11C2, C4.23(C2×C3⋊D4), C6.104(C2×C4○D4), (C2×C6).579(C2×D4), C2.68(C2×D4⋊2S3), C2.13(C22×C3⋊D4), (C2×C6).176(C4○D4), (C2×C6.D4)⋊25C2, (C2×C4).153(C3⋊D4), (C2×C4).623(C22×S3), C22.109(C2×C3⋊D4), SmallGroup(192,1356)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 744 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C6, C6 [×6], C6 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C2×C6 [×20], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×8], C24 [×2], Dic6 [×8], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×6], C3×D4 [×8], C22×C6, C22×C6 [×4], C22×C6 [×12], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, C4×Dic3 [×4], C6.D4 [×16], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C2×C4.4D4, C2×C4×Dic3, C23.12D6 [×8], C2×C6.D4 [×4], C22×Dic6, D4×C2×C6, C2×C23.12D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], D4⋊2S3 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C4.4D4, C23.12D6 [×4], C2×D4⋊2S3 [×2], C22×C3⋊D4, C2×C23.12D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
►On 96 points
Generators in S96
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 85)(11 86)(12 87)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 73)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 61)
(1 84)(2 79)(3 74)(4 81)(5 76)(6 83)(7 78)(8 73)(9 80)(10 75)(11 82)(12 77)(13 52)(14 59)(15 54)(16 49)(17 56)(18 51)(19 58)(20 53)(21 60)(22 55)(23 50)(24 57)(25 90)(26 85)(27 92)(28 87)(29 94)(30 89)(31 96)(32 91)(33 86)(34 93)(35 88)(36 95)(37 71)(38 66)(39 61)(40 68)(41 63)(42 70)(43 65)(44 72)(45 67)(46 62)(47 69)(48 64)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 25)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 38)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 85)
(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 21 32 41)(2 14 33 46)(3 19 34 39)(4 24 35 44)(5 17 36 37)(6 22 25 42)(7 15 26 47)(8 20 27 40)(9 13 28 45)(10 18 29 38)(11 23 30 43)(12 16 31 48)(49 83 64 90)(50 76 65 95)(51 81 66 88)(52 74 67 93)(53 79 68 86)(54 84 69 91)(55 77 70 96)(56 82 71 89)(57 75 72 94)(58 80 61 87)(59 73 62 92)(60 78 63 85)
G:=sub<Sym(96)| (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,61), (1,84)(2,79)(3,74)(4,81)(5,76)(6,83)(7,78)(8,73)(9,80)(10,75)(11,82)(12,77)(13,52)(14,59)(15,54)(16,49)(17,56)(18,51)(19,58)(20,53)(21,60)(22,55)(23,50)(24,57)(25,90)(26,85)(27,92)(28,87)(29,94)(30,89)(31,96)(32,91)(33,86)(34,93)(35,88)(36,95)(37,71)(38,66)(39,61)(40,68)(41,63)(42,70)(43,65)(44,72)(45,67)(46,62)(47,69)(48,64), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (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), (1,21,32,41)(2,14,33,46)(3,19,34,39)(4,24,35,44)(5,17,36,37)(6,22,25,42)(7,15,26,47)(8,20,27,40)(9,13,28,45)(10,18,29,38)(11,23,30,43)(12,16,31,48)(49,83,64,90)(50,76,65,95)(51,81,66,88)(52,74,67,93)(53,79,68,86)(54,84,69,91)(55,77,70,96)(56,82,71,89)(57,75,72,94)(58,80,61,87)(59,73,62,92)(60,78,63,85)>;
G:=Group( (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,61), (1,84)(2,79)(3,74)(4,81)(5,76)(6,83)(7,78)(8,73)(9,80)(10,75)(11,82)(12,77)(13,52)(14,59)(15,54)(16,49)(17,56)(18,51)(19,58)(20,53)(21,60)(22,55)(23,50)(24,57)(25,90)(26,85)(27,92)(28,87)(29,94)(30,89)(31,96)(32,91)(33,86)(34,93)(35,88)(36,95)(37,71)(38,66)(39,61)(40,68)(41,63)(42,70)(43,65)(44,72)(45,67)(46,62)(47,69)(48,64), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (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), (1,21,32,41)(2,14,33,46)(3,19,34,39)(4,24,35,44)(5,17,36,37)(6,22,25,42)(7,15,26,47)(8,20,27,40)(9,13,28,45)(10,18,29,38)(11,23,30,43)(12,16,31,48)(49,83,64,90)(50,76,65,95)(51,81,66,88)(52,74,67,93)(53,79,68,86)(54,84,69,91)(55,77,70,96)(56,82,71,89)(57,75,72,94)(58,80,61,87)(59,73,62,92)(60,78,63,85) );
G=PermutationGroup([(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,85),(11,86),(12,87),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,73),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,61)], [(1,84),(2,79),(3,74),(4,81),(5,76),(6,83),(7,78),(8,73),(9,80),(10,75),(11,82),(12,77),(13,52),(14,59),(15,54),(16,49),(17,56),(18,51),(19,58),(20,53),(21,60),(22,55),(23,50),(24,57),(25,90),(26,85),(27,92),(28,87),(29,94),(30,89),(31,96),(32,91),(33,86),(34,93),(35,88),(36,95),(37,71),(38,66),(39,61),(40,68),(41,63),(42,70),(43,65),(44,72),(45,67),(46,62),(47,69),(48,64)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,25),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,37),(24,38),(49,70),(50,71),(51,72),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,85)], [(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)], [(1,21,32,41),(2,14,33,46),(3,19,34,39),(4,24,35,44),(5,17,36,37),(6,22,25,42),(7,15,26,47),(8,20,27,40),(9,13,28,45),(10,18,29,38),(11,23,30,43),(12,16,31,48),(49,83,64,90),(50,76,65,95),(51,81,66,88),(52,74,67,93),(53,79,68,86),(54,84,69,91),(55,77,70,96),(56,82,71,89),(57,75,72,94),(58,80,61,87),(59,73,62,92),(60,78,63,85)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,12,0],[12,0,0,0,0,0,0,9,0,0,0,10,0,0,0,0,0,0,0,8,0,0,0,5,0] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D4⋊2S3 |
| kernel | C2×C23.12D6 | C2×C4×Dic3 | C23.12D6 | C2×C6.D4 | C22×Dic6 | D4×C2×C6 | C22×D4 | C2×C12 | C22×C4 | C2×D4 | C24 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 1 | 8 | 4 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{12}D_6 % in TeX
G:=Group("C2xC2^3.12D6"); // GroupNames label
G:=SmallGroup(192,1356);
// by ID
G=gap.SmallGroup(192,1356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,1571,185,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=d*c=c*d,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,f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀