direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C22⋊Q8, C4.63(C6×D4), (C22×C6)⋊6Q8, C22⋊2(C6×Q8), C23⋊4(C3×Q8), (C2×C12).524D4, C12.470(C2×D4), C24.37(C2×C6), (C23×C4).15C6, (C22×Q8)⋊11C6, (C6×Q8)⋊48C22, C22.60(C6×D4), C6.57(C22×Q8), (C23×C12).25C2, (C2×C6).343C24, C6.182(C22×D4), (C2×C12).656C23, C22.17(C23×C6), C23.74(C22×C6), (C23×C6).91C22, (C22×C6).258C23, (C22×C12).444C22, C2.6(D4×C2×C6), C2.3(Q8×C2×C6), (C6×C4⋊C4)⋊42C2, (C2×C4⋊C4)⋊15C6, (Q8×C2×C6)⋊15C2, (C2×C6)⋊5(C2×Q8), C4⋊C4⋊10(C2×C6), C2.6(C6×C4○D4), (C2×Q8)⋊10(C2×C6), (C3×C4⋊C4)⋊66C22, C6.225(C2×C4○D4), (C2×C6).682(C2×D4), (C2×C4).135(C3×D4), (C2×C22⋊C4).11C6, (C6×C22⋊C4).31C2, C22⋊C4.10(C2×C6), (C2×C4).12(C22×C6), (C22×C4).58(C2×C6), C22.30(C3×C4○D4), (C2×C6).230(C4○D4), (C3×C22⋊C4).144C22, SmallGroup(192,1412)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×16], C2×C4 [×18], Q8 [×8], C23, C23 [×6], C23 [×4], C12 [×4], C12 [×10], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24, C2×C12 [×16], C2×C12 [×18], C3×Q8 [×8], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, C3×C22⋊C4 [×8], C3×C4⋊C4 [×12], C22×C12 [×2], C22×C12 [×8], C22×C12 [×4], C6×Q8 [×4], C6×Q8 [×4], C23×C6, C2×C22⋊Q8, C6×C22⋊C4 [×2], C6×C4⋊C4, C6×C4⋊C4 [×2], C3×C22⋊Q8 [×8], C23×C12, Q8×C2×C6, C6×C22⋊Q8
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], Q8 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C3×D4 [×4], C3×Q8 [×4], C22×C6 [×15], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C6×D4 [×6], C6×Q8 [×6], C3×C4○D4 [×2], C23×C6, C2×C22⋊Q8, C3×C22⋊Q8 [×4], D4×C2×C6, Q8×C2×C6, C6×C4○D4, C6×C22⋊Q8
Generators and relations
G = < a,b,c,d,e | a6=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-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)
(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 96)(20 91)(21 92)(22 93)(23 94)(24 95)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 96)(20 91)(21 92)(22 93)(23 94)(24 95)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
(1 47 32 65)(2 48 33 66)(3 43 34 61)(4 44 35 62)(5 45 36 63)(6 46 31 64)(7 88 20 70)(8 89 21 71)(9 90 22 72)(10 85 23 67)(11 86 24 68)(12 87 19 69)(13 82 91 76)(14 83 92 77)(15 84 93 78)(16 79 94 73)(17 80 95 74)(18 81 96 75)(25 58 42 52)(26 59 37 53)(27 60 38 54)(28 55 39 49)(29 56 40 50)(30 57 41 51)
(1 71 32 89)(2 72 33 90)(3 67 34 85)(4 68 35 86)(5 69 36 87)(6 70 31 88)(7 46 20 64)(8 47 21 65)(9 48 22 66)(10 43 23 61)(11 44 24 62)(12 45 19 63)(13 52 91 58)(14 53 92 59)(15 54 93 60)(16 49 94 55)(17 50 95 56)(18 51 96 57)(25 82 42 76)(26 83 37 77)(27 84 38 78)(28 79 39 73)(29 80 40 74)(30 81 41 75)
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), (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,47,32,65)(2,48,33,66)(3,43,34,61)(4,44,35,62)(5,45,36,63)(6,46,31,64)(7,88,20,70)(8,89,21,71)(9,90,22,72)(10,85,23,67)(11,86,24,68)(12,87,19,69)(13,82,91,76)(14,83,92,77)(15,84,93,78)(16,79,94,73)(17,80,95,74)(18,81,96,75)(25,58,42,52)(26,59,37,53)(27,60,38,54)(28,55,39,49)(29,56,40,50)(30,57,41,51), (1,71,32,89)(2,72,33,90)(3,67,34,85)(4,68,35,86)(5,69,36,87)(6,70,31,88)(7,46,20,64)(8,47,21,65)(9,48,22,66)(10,43,23,61)(11,44,24,62)(12,45,19,63)(13,52,91,58)(14,53,92,59)(15,54,93,60)(16,49,94,55)(17,50,95,56)(18,51,96,57)(25,82,42,76)(26,83,37,77)(27,84,38,78)(28,79,39,73)(29,80,40,74)(30,81,41,75)>;
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), (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,47,32,65)(2,48,33,66)(3,43,34,61)(4,44,35,62)(5,45,36,63)(6,46,31,64)(7,88,20,70)(8,89,21,71)(9,90,22,72)(10,85,23,67)(11,86,24,68)(12,87,19,69)(13,82,91,76)(14,83,92,77)(15,84,93,78)(16,79,94,73)(17,80,95,74)(18,81,96,75)(25,58,42,52)(26,59,37,53)(27,60,38,54)(28,55,39,49)(29,56,40,50)(30,57,41,51), (1,71,32,89)(2,72,33,90)(3,67,34,85)(4,68,35,86)(5,69,36,87)(6,70,31,88)(7,46,20,64)(8,47,21,65)(9,48,22,66)(10,43,23,61)(11,44,24,62)(12,45,19,63)(13,52,91,58)(14,53,92,59)(15,54,93,60)(16,49,94,55)(17,50,95,56)(18,51,96,57)(25,82,42,76)(26,83,37,77)(27,84,38,78)(28,79,39,73)(29,80,40,74)(30,81,41,75) );
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)], [(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(19,96),(20,91),(21,92),(22,93),(23,94),(24,95),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(19,96),(20,91),(21,92),(22,93),(23,94),(24,95),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90)], [(1,47,32,65),(2,48,33,66),(3,43,34,61),(4,44,35,62),(5,45,36,63),(6,46,31,64),(7,88,20,70),(8,89,21,71),(9,90,22,72),(10,85,23,67),(11,86,24,68),(12,87,19,69),(13,82,91,76),(14,83,92,77),(15,84,93,78),(16,79,94,73),(17,80,95,74),(18,81,96,75),(25,58,42,52),(26,59,37,53),(27,60,38,54),(28,55,39,49),(29,56,40,50),(30,57,41,51)], [(1,71,32,89),(2,72,33,90),(3,67,34,85),(4,68,35,86),(5,69,36,87),(6,70,31,88),(7,46,20,64),(8,47,21,65),(9,48,22,66),(10,43,23,61),(11,44,24,62),(12,45,19,63),(13,52,91,58),(14,53,92,59),(15,54,93,60),(16,49,94,55),(17,50,95,56),(18,51,96,57),(25,82,42,76),(26,83,37,77),(27,84,38,78),(28,79,39,73),(29,80,40,74),(30,81,41,75)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[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],[12,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,12,0] >;
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12P | 12Q | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | Q8 | C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 |
| kernel | C6×C22⋊Q8 | C6×C22⋊C4 | C6×C4⋊C4 | C3×C22⋊Q8 | C23×C12 | Q8×C2×C6 | C2×C22⋊Q8 | C2×C22⋊C4 | C2×C4⋊C4 | C22⋊Q8 | C23×C4 | C22×Q8 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 8 | 1 | 1 | 2 | 4 | 6 | 16 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_6\times C_2^2\rtimes Q_8
% in TeX
G:=Group("C6xC2^2:Q8"); // GroupNames label
G:=SmallGroup(192,1412);
// by ID
G=gap.SmallGroup(192,1412);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,344,2102]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations