direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C22.D4, (C23×C4)⋊8C6, (C23×C12)⋊7C2, C24.38(C2×C6), C22.61(C6×D4), C23.54(C3×D4), (C2×C6).344C24, (C22×D4).11C6, C6.183(C22×D4), (C22×C6).171D4, C23.5(C22×C6), (C2×C12).657C23, (C22×C12)⋊59C22, (C6×D4).316C22, C22.18(C23×C6), (C23×C6).92C22, (C22×C6).259C23, C2.7(D4×C2×C6), (C6×C4⋊C4)⋊43C2, (C2×C4⋊C4)⋊16C6, C4⋊C4⋊11(C2×C6), (D4×C2×C6).23C2, C2.7(C6×C4○D4), (C2×C22⋊C4)⋊10C6, (C6×C22⋊C4)⋊30C2, (C3×C4⋊C4)⋊67C22, C22⋊C4⋊12(C2×C6), (C22×C4)⋊19(C2×C6), (C2×D4).61(C2×C6), C6.226(C2×C4○D4), (C2×C6).415(C2×D4), (C2×C4).13(C22×C6), C22.31(C3×C4○D4), (C2×C6).231(C4○D4), (C3×C22⋊C4)⋊66C22, SmallGroup(192,1413)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 530 in 342 conjugacy classes, 178 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×6], C3, C4 [×10], C22, C22 [×10], C22 [×22], C6, C6 [×6], C6 [×6], C2×C4 [×10], C2×C4 [×18], D4 [×8], C23, C23 [×8], C23 [×10], C12 [×10], C2×C6, C2×C6 [×10], C2×C6 [×22], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×8], C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×C12 [×10], C2×C12 [×18], C3×D4 [×8], C22×C6, C22×C6 [×8], C22×C6 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C3×C22⋊C4 [×12], C3×C4⋊C4 [×8], C22×C12, C22×C12 [×8], C22×C12 [×4], C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C2×C22.D4, C6×C22⋊C4, C6×C22⋊C4 [×2], C6×C4⋊C4 [×2], C3×C22.D4 [×8], C23×C12, D4×C2×C6, C6×C22.D4
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C4○D4 [×4], C24, C3×D4 [×4], C22×C6 [×15], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C6×D4 [×6], C3×C4○D4 [×4], C23×C6, C2×C22.D4, C3×C22.D4 [×4], D4×C2×C6, C6×C4○D4 [×2], C6×C22.D4
Generators and relations
G = < a,b,c,d,e | a6=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
►On 96 points
(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 83)(2 84)(3 79)(4 80)(5 81)(6 82)(7 52)(8 53)(9 54)(10 49)(11 50)(12 51)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 57)(20 58)(21 59)(22 60)(23 55)(24 56)(25 70)(26 71)(27 72)(28 67)(29 68)(30 69)(31 76)(32 77)(33 78)(34 73)(35 74)(36 75)(37 89)(38 90)(39 85)(40 86)(41 87)(42 88)(61 94)(62 95)(63 96)(64 91)(65 92)(66 93)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)
(1 56 26 44)(2 57 27 45)(3 58 28 46)(4 59 29 47)(5 60 30 48)(6 55 25 43)(7 67 91 79)(8 68 92 80)(9 69 93 81)(10 70 94 82)(11 71 95 83)(12 72 96 84)(13 73 20 85)(14 74 21 86)(15 75 22 87)(16 76 23 88)(17 77 24 89)(18 78 19 90)(31 49 42 61)(32 50 37 62)(33 51 38 63)(34 52 39 64)(35 53 40 65)(36 54 41 66)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)(43 64)(44 65)(45 66)(46 61)(47 62)(48 63)(49 58)(50 59)(51 60)(52 55)(53 56)(54 57)(67 76)(68 77)(69 78)(70 73)(71 74)(72 75)(79 88)(80 89)(81 90)(82 85)(83 86)(84 87)(91 94)(92 95)(93 96)
G:=sub<Sym(96)| (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), (1,83)(2,84)(3,79)(4,80)(5,81)(6,82)(7,52)(8,53)(9,54)(10,49)(11,50)(12,51)(13,46)(14,47)(15,48)(16,43)(17,44)(18,45)(19,57)(20,58)(21,59)(22,60)(23,55)(24,56)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,76)(32,77)(33,78)(34,73)(35,74)(36,75)(37,89)(38,90)(39,85)(40,86)(41,87)(42,88)(61,94)(62,95)(63,96)(64,91)(65,92)(66,93), (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (1,56,26,44)(2,57,27,45)(3,58,28,46)(4,59,29,47)(5,60,30,48)(6,55,25,43)(7,67,91,79)(8,68,92,80)(9,69,93,81)(10,70,94,82)(11,71,95,83)(12,72,96,84)(13,73,20,85)(14,74,21,86)(15,75,22,87)(16,76,23,88)(17,77,24,89)(18,78,19,90)(31,49,42,61)(32,50,37,62)(33,51,38,63)(34,52,39,64)(35,53,40,65)(36,54,41,66), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38)(43,64)(44,65)(45,66)(46,61)(47,62)(48,63)(49,58)(50,59)(51,60)(52,55)(53,56)(54,57)(67,76)(68,77)(69,78)(70,73)(71,74)(72,75)(79,88)(80,89)(81,90)(82,85)(83,86)(84,87)(91,94)(92,95)(93,96)>;
G:=Group( (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), (1,83)(2,84)(3,79)(4,80)(5,81)(6,82)(7,52)(8,53)(9,54)(10,49)(11,50)(12,51)(13,46)(14,47)(15,48)(16,43)(17,44)(18,45)(19,57)(20,58)(21,59)(22,60)(23,55)(24,56)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,76)(32,77)(33,78)(34,73)(35,74)(36,75)(37,89)(38,90)(39,85)(40,86)(41,87)(42,88)(61,94)(62,95)(63,96)(64,91)(65,92)(66,93), (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (1,56,26,44)(2,57,27,45)(3,58,28,46)(4,59,29,47)(5,60,30,48)(6,55,25,43)(7,67,91,79)(8,68,92,80)(9,69,93,81)(10,70,94,82)(11,71,95,83)(12,72,96,84)(13,73,20,85)(14,74,21,86)(15,75,22,87)(16,76,23,88)(17,77,24,89)(18,78,19,90)(31,49,42,61)(32,50,37,62)(33,51,38,63)(34,52,39,64)(35,53,40,65)(36,54,41,66), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38)(43,64)(44,65)(45,66)(46,61)(47,62)(48,63)(49,58)(50,59)(51,60)(52,55)(53,56)(54,57)(67,76)(68,77)(69,78)(70,73)(71,74)(72,75)(79,88)(80,89)(81,90)(82,85)(83,86)(84,87)(91,94)(92,95)(93,96) );
G=PermutationGroup([(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)], [(1,83),(2,84),(3,79),(4,80),(5,81),(6,82),(7,52),(8,53),(9,54),(10,49),(11,50),(12,51),(13,46),(14,47),(15,48),(16,43),(17,44),(18,45),(19,57),(20,58),(21,59),(22,60),(23,55),(24,56),(25,70),(26,71),(27,72),(28,67),(29,68),(30,69),(31,76),(32,77),(33,78),(34,73),(35,74),(36,75),(37,89),(38,90),(39,85),(40,86),(41,87),(42,88),(61,94),(62,95),(63,96),(64,91),(65,92),(66,93)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,31),(7,20),(8,21),(9,22),(10,23),(11,24),(12,19),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(25,42),(26,37),(27,38),(28,39),(29,40),(30,41),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84)], [(1,56,26,44),(2,57,27,45),(3,58,28,46),(4,59,29,47),(5,60,30,48),(6,55,25,43),(7,67,91,79),(8,68,92,80),(9,69,93,81),(10,70,94,82),(11,71,95,83),(12,72,96,84),(13,73,20,85),(14,74,21,86),(15,75,22,87),(16,76,23,88),(17,77,24,89),(18,78,19,90),(31,49,42,61),(32,50,37,62),(33,51,38,63),(34,52,39,64),(35,53,40,65),(36,54,41,66)], [(1,29),(2,30),(3,25),(4,26),(5,27),(6,28),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(31,39),(32,40),(33,41),(34,42),(35,37),(36,38),(43,64),(44,65),(45,66),(46,61),(47,62),(48,63),(49,58),(50,59),(51,60),(52,55),(53,56),(54,57),(67,76),(68,77),(69,78),(70,73),(71,74),(72,75),(79,88),(80,89),(81,90),(82,85),(83,86),(84,87),(91,94),(92,95),(93,96)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 10 | 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 | 0 | 5 |
| 0 | 0 | 0 | 8 | 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 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,5,0],[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],[12,0,0,0,0,0,1,12,0,0,0,2,12,0,0,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1] >;
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4N | 6A | ··· | 6N | 6O | ··· | 6V | 6W | 6X | 6Y | 6Z | 12A | ··· | 12P | 12Q | ··· | 12AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | C6×C22.D4 | C6×C22⋊C4 | C6×C4⋊C4 | C3×C22.D4 | C23×C12 | D4×C2×C6 | C2×C22.D4 | C2×C22⋊C4 | C2×C4⋊C4 | C22.D4 | C23×C4 | C22×D4 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 3 | 2 | 8 | 1 | 1 | 2 | 6 | 4 | 16 | 2 | 2 | 4 | 8 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_6\times C_2^2.D_4
% in TeX
G:=Group("C6xC2^2.D4"); // GroupNames label
G:=SmallGroup(192,1413);
// by ID
G=gap.SmallGroup(192,1413);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,268]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations