direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×D4×Q8, C6.1192- (1+4), C4⋊2(C6×Q8), C4⋊Q8⋊14C6, (C4×Q8)⋊16C6, C12⋊10(C2×Q8), C4.43(C6×D4), C22⋊3(C6×Q8), (Q8×C12)⋊32C2, (C4×D4).10C6, C22⋊Q8⋊14C6, (D4×C12).25C2, C12.404(C2×D4), C42.44(C2×C6), (C22×Q8)⋊15C6, C6.63(C22×Q8), (C2×C6).369C24, C6.197(C22×D4), (C2×C12).676C23, (C4×C12).285C22, (C6×D4).334C22, C22.43(C23×C6), C23.47(C22×C6), (C6×Q8).275C22, (C22×C6).265C23, C2.11(C3×2- (1+4)), (C22×C12).455C22, C2.9(Q8×C2×C6), (Q8×C2×C6)⋊19C2, (C2×C6)⋊8(C2×Q8), C2.21(D4×C2×C6), (C3×C4⋊Q8)⋊35C2, C4⋊C4.32(C2×C6), (C2×D4).80(C2×C6), (C3×C22⋊Q8)⋊41C2, (C2×Q8).74(C2×C6), C22⋊C4.20(C2×C6), (C22×C4).72(C2×C6), (C2×C4).34(C22×C6), (C3×C4⋊C4).397C22, (C3×C22⋊C4).153C22, SmallGroup(192,1438)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 280 conjugacy classes, 182 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C4 [×9], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4, C2×C4 [×12], C2×C4 [×12], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], C12 [×8], C12 [×9], C2×C6, C2×C6 [×4], C2×C6 [×4], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×8], C2×C12, C2×C12 [×12], C2×C12 [×12], C3×D4 [×4], C3×Q8 [×4], C3×Q8 [×12], C22×C6 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], C4×C12 [×3], C3×C22⋊C4 [×6], C3×C4⋊C4 [×12], C22×C12 [×6], C6×D4, C6×Q8, C6×Q8 [×6], C6×Q8 [×8], D4×Q8, D4×C12 [×3], Q8×C12, C3×C22⋊Q8 [×6], C3×C4⋊Q8 [×3], Q8×C2×C6 [×2], C3×D4×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], C24, C3×D4 [×4], C3×Q8 [×4], C22×C6 [×15], C22×D4, C22×Q8, 2- (1+4), C6×D4 [×6], C6×Q8 [×6], C23×C6, D4×Q8, D4×C2×C6, Q8×C2×C6, C3×2- (1+4), C3×D4×Q8
Generators and relations
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 96 points
(1 27 83)(2 28 84)(3 25 81)(4 26 82)(5 51 23)(6 52 24)(7 49 21)(8 50 22)(9 37 71)(10 38 72)(11 39 69)(12 40 70)(13 57 19)(14 58 20)(15 59 17)(16 60 18)(29 85 89)(30 86 90)(31 87 91)(32 88 92)(33 61 93)(34 62 94)(35 63 95)(36 64 96)(41 75 79)(42 76 80)(43 73 77)(44 74 78)(45 67 56)(46 68 53)(47 65 54)(48 66 55)
(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 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 38)(39 40)(41 44)(42 43)(45 48)(46 47)(49 52)(50 51)(53 54)(55 56)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 70)(71 72)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)
(1 75 13 23)(2 76 14 24)(3 73 15 21)(4 74 16 22)(5 27 79 57)(6 28 80 58)(7 25 77 59)(8 26 78 60)(9 68 88 36)(10 65 85 33)(11 66 86 34)(12 67 87 35)(17 49 81 43)(18 50 82 44)(19 51 83 41)(20 52 84 42)(29 93 72 47)(30 94 69 48)(31 95 70 45)(32 96 71 46)(37 53 92 64)(38 54 89 61)(39 55 90 62)(40 56 91 63)
(1 35 13 67)(2 36 14 68)(3 33 15 65)(4 34 16 66)(5 40 79 91)(6 37 80 92)(7 38 77 89)(8 39 78 90)(9 76 88 24)(10 73 85 21)(11 74 86 22)(12 75 87 23)(17 47 81 93)(18 48 82 94)(19 45 83 95)(20 46 84 96)(25 61 59 54)(26 62 60 55)(27 63 57 56)(28 64 58 53)(29 49 72 43)(30 50 69 44)(31 51 70 41)(32 52 71 42)
G:=sub<Sym(96)| (1,27,83)(2,28,84)(3,25,81)(4,26,82)(5,51,23)(6,52,24)(7,49,21)(8,50,22)(9,37,71)(10,38,72)(11,39,69)(12,40,70)(13,57,19)(14,58,20)(15,59,17)(16,60,18)(29,85,89)(30,86,90)(31,87,91)(32,88,92)(33,61,93)(34,62,94)(35,63,95)(36,64,96)(41,75,79)(42,76,80)(43,73,77)(44,74,78)(45,67,56)(46,68,53)(47,65,54)(48,66,55), (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), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)(45,48)(46,47)(49,52)(50,51)(53,54)(55,56)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,70)(71,72)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95), (1,75,13,23)(2,76,14,24)(3,73,15,21)(4,74,16,22)(5,27,79,57)(6,28,80,58)(7,25,77,59)(8,26,78,60)(9,68,88,36)(10,65,85,33)(11,66,86,34)(12,67,87,35)(17,49,81,43)(18,50,82,44)(19,51,83,41)(20,52,84,42)(29,93,72,47)(30,94,69,48)(31,95,70,45)(32,96,71,46)(37,53,92,64)(38,54,89,61)(39,55,90,62)(40,56,91,63), (1,35,13,67)(2,36,14,68)(3,33,15,65)(4,34,16,66)(5,40,79,91)(6,37,80,92)(7,38,77,89)(8,39,78,90)(9,76,88,24)(10,73,85,21)(11,74,86,22)(12,75,87,23)(17,47,81,93)(18,48,82,94)(19,45,83,95)(20,46,84,96)(25,61,59,54)(26,62,60,55)(27,63,57,56)(28,64,58,53)(29,49,72,43)(30,50,69,44)(31,51,70,41)(32,52,71,42)>;
G:=Group( (1,27,83)(2,28,84)(3,25,81)(4,26,82)(5,51,23)(6,52,24)(7,49,21)(8,50,22)(9,37,71)(10,38,72)(11,39,69)(12,40,70)(13,57,19)(14,58,20)(15,59,17)(16,60,18)(29,85,89)(30,86,90)(31,87,91)(32,88,92)(33,61,93)(34,62,94)(35,63,95)(36,64,96)(41,75,79)(42,76,80)(43,73,77)(44,74,78)(45,67,56)(46,68,53)(47,65,54)(48,66,55), (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), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)(45,48)(46,47)(49,52)(50,51)(53,54)(55,56)(57,60)(58,59)(61,64)(62,63)(65,68)(66,67)(69,70)(71,72)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95), (1,75,13,23)(2,76,14,24)(3,73,15,21)(4,74,16,22)(5,27,79,57)(6,28,80,58)(7,25,77,59)(8,26,78,60)(9,68,88,36)(10,65,85,33)(11,66,86,34)(12,67,87,35)(17,49,81,43)(18,50,82,44)(19,51,83,41)(20,52,84,42)(29,93,72,47)(30,94,69,48)(31,95,70,45)(32,96,71,46)(37,53,92,64)(38,54,89,61)(39,55,90,62)(40,56,91,63), (1,35,13,67)(2,36,14,68)(3,33,15,65)(4,34,16,66)(5,40,79,91)(6,37,80,92)(7,38,77,89)(8,39,78,90)(9,76,88,24)(10,73,85,21)(11,74,86,22)(12,75,87,23)(17,47,81,93)(18,48,82,94)(19,45,83,95)(20,46,84,96)(25,61,59,54)(26,62,60,55)(27,63,57,56)(28,64,58,53)(29,49,72,43)(30,50,69,44)(31,51,70,41)(32,52,71,42) );
G=PermutationGroup([(1,27,83),(2,28,84),(3,25,81),(4,26,82),(5,51,23),(6,52,24),(7,49,21),(8,50,22),(9,37,71),(10,38,72),(11,39,69),(12,40,70),(13,57,19),(14,58,20),(15,59,17),(16,60,18),(29,85,89),(30,86,90),(31,87,91),(32,88,92),(33,61,93),(34,62,94),(35,63,95),(36,64,96),(41,75,79),(42,76,80),(43,73,77),(44,74,78),(45,67,56),(46,68,53),(47,65,54),(48,66,55)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35),(37,38),(39,40),(41,44),(42,43),(45,48),(46,47),(49,52),(50,51),(53,54),(55,56),(57,60),(58,59),(61,64),(62,63),(65,68),(66,67),(69,70),(71,72),(73,76),(74,75),(77,80),(78,79),(81,84),(82,83),(85,88),(86,87),(89,92),(90,91),(93,96),(94,95)], [(1,75,13,23),(2,76,14,24),(3,73,15,21),(4,74,16,22),(5,27,79,57),(6,28,80,58),(7,25,77,59),(8,26,78,60),(9,68,88,36),(10,65,85,33),(11,66,86,34),(12,67,87,35),(17,49,81,43),(18,50,82,44),(19,51,83,41),(20,52,84,42),(29,93,72,47),(30,94,69,48),(31,95,70,45),(32,96,71,46),(37,53,92,64),(38,54,89,61),(39,55,90,62),(40,56,91,63)], [(1,35,13,67),(2,36,14,68),(3,33,15,65),(4,34,16,66),(5,40,79,91),(6,37,80,92),(7,38,77,89),(8,39,78,90),(9,76,88,24),(10,73,85,21),(11,74,86,22),(12,75,87,23),(17,47,81,93),(18,48,82,94),(19,45,83,95),(20,46,84,96),(25,61,59,54),(26,62,60,55),(27,63,57,56),(28,64,58,53),(29,49,72,43),(30,50,69,44),(31,51,70,41),(32,52,71,42)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 11 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,12,0],[8,5,0,0,0,5,0,0,0,0,12,0,0,0,0,12],[12,1,0,0,11,1,0,0,0,0,12,0,0,0,0,12] >;
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4Q | 6A | ··· | 6F | 6G | ··· | 6N | 12A | ··· | 12P | 12Q | ··· | 12AH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | Q8 | D4 | C3×Q8 | C3×D4 | 2- (1+4) | C3×2- (1+4) |
| kernel | C3×D4×Q8 | D4×C12 | Q8×C12 | C3×C22⋊Q8 | C3×C4⋊Q8 | Q8×C2×C6 | D4×Q8 | C4×D4 | C4×Q8 | C22⋊Q8 | C4⋊Q8 | C22×Q8 | C3×D4 | C3×Q8 | D4 | Q8 | C6 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 2 | 6 | 2 | 12 | 6 | 4 | 4 | 4 | 8 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_3\times D_4\times Q_8
% in TeX
G:=Group("C3xD4xQ8"); // GroupNames label
G:=SmallGroup(192,1438);
// by ID
G=gap.SmallGroup(192,1438);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,344,2102,794,192]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=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,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations