direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.11D6, C12.31C24, D12.28C23, Dic6.27C23, (C2×Q8)⋊30D6, C3⋊C8.13C23, (C2×C12).211D4, C12.255(C2×D4), C6⋊4(C8.C22), C4.31(S3×C23), (C6×Q8)⋊34C22, (C22×Q8)⋊11S3, C3⋊Q16⋊15C22, (C22×C6).210D4, C6.150(C22×D4), (C22×C4).290D6, (C3×Q8).20C23, Q8.30(C22×S3), (C2×C12).548C23, Q8⋊2S3⋊16C22, C4○D12.57C22, C4.Dic3⋊33C22, (C2×D12).277C22, C23.100(C3⋊D4), (C22×C12).280C22, (C2×Dic6).305C22, (Q8×C2×C6)⋊3C2, C3⋊5(C2×C8.C22), C4.25(C2×C3⋊D4), (C2×C3⋊Q16)⋊30C2, (C2×C6).585(C2×D4), (C2×Q8⋊2S3)⋊30C2, (C2×C4○D12).24C2, (C2×C4).93(C3⋊D4), (C2×C3⋊C8).183C22, (C2×C4.Dic3)⋊27C2, C2.23(C22×C3⋊D4), (C2×C4).240(C22×S3), C22.113(C2×C3⋊D4), SmallGroup(192,1367)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 584 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×4], Q8 [×9], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×6], C22×S3, C22×C6, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], Q8⋊2S3 [×8], C3⋊Q16 [×8], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C22×C12, C6×Q8 [×6], C6×Q8 [×3], C2×C8.C22, C2×C4.Dic3, C2×Q8⋊2S3 [×2], Q8.11D6 [×8], C2×C3⋊Q16 [×2], C2×C4○D12, Q8×C2×C6, C2×Q8.11D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8.C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8.C22, Q8.11D6 [×2], C22×C3⋊D4, C2×Q8.11D6
Generators and relations
G = < a,b,c,d,e | a2=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d5 >
►On 96 points
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 71)(38 72)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(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 26 7 32)(2 27 8 33)(3 28 9 34)(4 29 10 35)(5 30 11 36)(6 31 12 25)(13 53 19 59)(14 54 20 60)(15 55 21 49)(16 56 22 50)(17 57 23 51)(18 58 24 52)(37 88 43 94)(38 89 44 95)(39 90 45 96)(40 91 46 85)(41 92 47 86)(42 93 48 87)(61 77 67 83)(62 78 68 84)(63 79 69 73)(64 80 70 74)(65 81 71 75)(66 82 72 76)
(1 77 7 83)(2 84 8 78)(3 79 9 73)(4 74 10 80)(5 81 11 75)(6 76 12 82)(13 86 19 92)(14 93 20 87)(15 88 21 94)(16 95 22 89)(17 90 23 96)(18 85 24 91)(25 66 31 72)(26 61 32 67)(27 68 33 62)(28 63 34 69)(29 70 35 64)(30 65 36 71)(37 49 43 55)(38 56 44 50)(39 51 45 57)(40 58 46 52)(41 53 47 59)(42 60 48 54)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 14 19 20)(15 24 21 18)(16 17 22 23)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 91 43 85)(38 96 44 90)(39 89 45 95)(40 94 46 88)(41 87 47 93)(42 92 48 86)(49 52 55 58)(50 57 56 51)(53 60 59 54)(61 76 67 82)(62 81 68 75)(63 74 69 80)(64 79 70 73)(65 84 71 78)(66 77 72 83)
G:=sub<Sym(96)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (1,26,7,32)(2,27,8,33)(3,28,9,34)(4,29,10,35)(5,30,11,36)(6,31,12,25)(13,53,19,59)(14,54,20,60)(15,55,21,49)(16,56,22,50)(17,57,23,51)(18,58,24,52)(37,88,43,94)(38,89,44,95)(39,90,45,96)(40,91,46,85)(41,92,47,86)(42,93,48,87)(61,77,67,83)(62,78,68,84)(63,79,69,73)(64,80,70,74)(65,81,71,75)(66,82,72,76), (1,77,7,83)(2,84,8,78)(3,79,9,73)(4,74,10,80)(5,81,11,75)(6,76,12,82)(13,86,19,92)(14,93,20,87)(15,88,21,94)(16,95,22,89)(17,90,23,96)(18,85,24,91)(25,66,31,72)(26,61,32,67)(27,68,33,62)(28,63,34,69)(29,70,35,64)(30,65,36,71)(37,49,43,55)(38,56,44,50)(39,51,45,57)(40,58,46,52)(41,53,47,59)(42,60,48,54), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,91,43,85)(38,96,44,90)(39,89,45,95)(40,94,46,88)(41,87,47,93)(42,92,48,86)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,76,67,82)(62,81,68,75)(63,74,69,80)(64,79,70,73)(65,84,71,78)(66,77,72,83)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,85), (1,26,7,32)(2,27,8,33)(3,28,9,34)(4,29,10,35)(5,30,11,36)(6,31,12,25)(13,53,19,59)(14,54,20,60)(15,55,21,49)(16,56,22,50)(17,57,23,51)(18,58,24,52)(37,88,43,94)(38,89,44,95)(39,90,45,96)(40,91,46,85)(41,92,47,86)(42,93,48,87)(61,77,67,83)(62,78,68,84)(63,79,69,73)(64,80,70,74)(65,81,71,75)(66,82,72,76), (1,77,7,83)(2,84,8,78)(3,79,9,73)(4,74,10,80)(5,81,11,75)(6,76,12,82)(13,86,19,92)(14,93,20,87)(15,88,21,94)(16,95,22,89)(17,90,23,96)(18,85,24,91)(25,66,31,72)(26,61,32,67)(27,68,33,62)(28,63,34,69)(29,70,35,64)(30,65,36,71)(37,49,43,55)(38,56,44,50)(39,51,45,57)(40,58,46,52)(41,53,47,59)(42,60,48,54), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,91,43,85)(38,96,44,90)(39,89,45,95)(40,94,46,88)(41,87,47,93)(42,92,48,86)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,76,67,82)(62,81,68,75)(63,74,69,80)(64,79,70,73)(65,84,71,78)(66,77,72,83) );
G=PermutationGroup([(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,13),(10,14),(11,15),(12,16),(25,56),(26,57),(27,58),(28,59),(29,60),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,71),(38,72),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,85)], [(1,26,7,32),(2,27,8,33),(3,28,9,34),(4,29,10,35),(5,30,11,36),(6,31,12,25),(13,53,19,59),(14,54,20,60),(15,55,21,49),(16,56,22,50),(17,57,23,51),(18,58,24,52),(37,88,43,94),(38,89,44,95),(39,90,45,96),(40,91,46,85),(41,92,47,86),(42,93,48,87),(61,77,67,83),(62,78,68,84),(63,79,69,73),(64,80,70,74),(65,81,71,75),(66,82,72,76)], [(1,77,7,83),(2,84,8,78),(3,79,9,73),(4,74,10,80),(5,81,11,75),(6,76,12,82),(13,86,19,92),(14,93,20,87),(15,88,21,94),(16,95,22,89),(17,90,23,96),(18,85,24,91),(25,66,31,72),(26,61,32,67),(27,68,33,62),(28,63,34,69),(29,70,35,64),(30,65,36,71),(37,49,43,55),(38,56,44,50),(39,51,45,57),(40,58,46,52),(41,53,47,59),(42,60,48,54)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,14,19,20),(15,24,21,18),(16,17,22,23),(25,32,31,26),(27,30,33,36),(28,35,34,29),(37,91,43,85),(38,96,44,90),(39,89,45,95),(40,94,46,88),(41,87,47,93),(42,92,48,86),(49,52,55,58),(50,57,56,51),(53,60,59,54),(61,76,67,82),(62,81,68,75),(63,74,69,80),(64,79,70,73),(65,84,71,78),(66,77,72,83)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 72 | 71 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 16 | 5 | 57 |
| 0 | 0 | 57 | 5 | 16 | 62 |
| 0 | 0 | 5 | 57 | 11 | 57 |
| 0 | 0 | 16 | 62 | 16 | 68 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 0 | 0 | 43 | 60 |
| 0 | 0 | 43 | 43 | 0 | 0 |
| 0 | 0 | 30 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 60 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 30 | 30 | 0 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,62,57,5,16,0,0,16,5,57,62,0,0,5,16,11,16,0,0,57,62,57,68],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,43,30,0,0,0,0,43,13,0,0,30,43,0,0,0,0,30,60,0,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,43,30,0,0,0,0,60,30,0,0,43,30,0,0,0,0,60,30,0,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8.C22 | Q8.11D6 |
| kernel | C2×Q8.11D6 | C2×C4.Dic3 | C2×Q8⋊2S3 | Q8.11D6 | C2×C3⋊Q16 | C2×C4○D12 | Q8×C2×C6 | C22×Q8 | C2×C12 | C22×C6 | C22×C4 | C2×Q8 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 1 | 3 | 1 | 1 | 6 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_8._{11}D_6 % in TeX
G:=Group("C2xQ8.11D6"); // GroupNames label
G:=SmallGroup(192,1367);
// by ID
G=gap.SmallGroup(192,1367);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,136,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^5>;
// generators/relations