direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.11D6, C24.39D6, C22⋊C4⋊41D6, D6⋊C4⋊47C22, C6⋊2(C4.4D4), (C2×C6).35C24, Dic3.1(C2×D4), C6.38(C22×D4), (C22×Dic6)⋊6C2, (C22×C4).330D6, C22.128(S3×D4), (C2×C12).574C23, (C2×Dic3).119D4, (C2×Dic6)⋊49C22, (C4×Dic3)⋊74C22, (C22×S3).7C23, C23.91(C22×S3), (C23×C6).61C22, C22.74(S3×C23), C22.74(C4○D12), C6.D4⋊46C22, (S3×C23).32C22, (C22×C6).388C23, C22.68(D4⋊2S3), (C22×C12).354C22, (C2×Dic3).181C23, (C22×Dic3).79C22, C2.12(C2×S3×D4), C3⋊2(C2×C4.4D4), (C2×D6⋊C4)⋊18C2, (C2×C4×Dic3)⋊31C2, C6.15(C2×C4○D4), (C2×C22⋊C4)⋊14S3, (C6×C22⋊C4)⋊19C2, C2.17(C2×C4○D12), (C2×C6).384(C2×D4), C2.10(C2×D4⋊2S3), (C2×C6).103(C4○D4), (C2×C6.D4)⋊17C2, (C3×C22⋊C4)⋊54C22, (C2×C4).260(C22×S3), (C2×C3⋊D4).90C22, (C22×C3⋊D4).11C2, SmallGroup(192,1050)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 872 in 330 conjugacy classes, 119 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×12], C22, C22 [×6], C22 [×20], S3 [×2], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×8], Q8 [×8], C23, C23 [×2], C23 [×14], Dic3 [×4], Dic3 [×4], C12 [×4], D6 [×10], C2×C6, C2×C6 [×6], C2×C6 [×10], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×12], C22×C4 [×2], C22×C4 [×3], C2×D4 [×8], C2×Q8 [×8], C24, C24, Dic6 [×8], C2×Dic3 [×10], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×4], C2×C12 [×4], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×2], C22×C6 [×6], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×3], C4.4D4 [×8], C22×D4, C22×Q8, C4×Dic3 [×4], D6⋊C4 [×8], C6.D4 [×4], C3×C22⋊C4 [×4], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×3], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, C2×C4.4D4, C23.11D6 [×8], C2×C4×Dic3, C2×D6⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, C22×Dic6, C22×C3⋊D4, C2×C23.11D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C4○D12 [×2], S3×D4 [×2], D4⋊2S3 [×2], S3×C23, C2×C4.4D4, C23.11D6 [×4], C2×C4○D12, C2×S3×D4, C2×D4⋊2S3, C2×C23.11D6
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, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >
►On 96 points
Generators in S96
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(61 94)(62 95)(63 96)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 49)(2 82)(3 51)(4 84)(5 53)(6 74)(7 55)(8 76)(9 57)(10 78)(11 59)(12 80)(13 40)(14 94)(15 42)(16 96)(17 44)(18 86)(19 46)(20 88)(21 48)(22 90)(23 38)(24 92)(25 37)(26 91)(27 39)(28 93)(29 41)(30 95)(31 43)(32 85)(33 45)(34 87)(35 47)(36 89)(50 70)(52 72)(54 62)(56 64)(58 66)(60 68)(61 73)(63 75)(65 77)(67 79)(69 81)(71 83)
(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 69)(2 70)(3 71)(4 72)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 85)(45 86)(46 87)(47 88)(48 89)(49 81)(50 82)(51 83)(52 84)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(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 63 68)(2 67 64 5)(3 4 65 66)(7 12 69 62)(8 61 70 11)(9 10 71 72)(13 24 34 33)(14 32 35 23)(15 22 36 31)(16 30 25 21)(17 20 26 29)(18 28 27 19)(37 42 96 89)(38 88 85 41)(39 40 86 87)(43 48 90 95)(44 94 91 47)(45 46 92 93)(49 80 75 54)(50 53 76 79)(51 78 77 52)(55 74 81 60)(56 59 82 73)(57 84 83 58)
G:=sub<Sym(96)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,49)(2,82)(3,51)(4,84)(5,53)(6,74)(7,55)(8,76)(9,57)(10,78)(11,59)(12,80)(13,40)(14,94)(15,42)(16,96)(17,44)(18,86)(19,46)(20,88)(21,48)(22,90)(23,38)(24,92)(25,37)(26,91)(27,39)(28,93)(29,41)(30,95)(31,43)(32,85)(33,45)(34,87)(35,47)(36,89)(50,70)(52,72)(54,62)(56,64)(58,66)(60,68)(61,73)(63,75)(65,77)(67,79)(69,81)(71,83), (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), (1,69)(2,70)(3,71)(4,72)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,85)(45,86)(46,87)(47,88)(48,89)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,6,63,68)(2,67,64,5)(3,4,65,66)(7,12,69,62)(8,61,70,11)(9,10,71,72)(13,24,34,33)(14,32,35,23)(15,22,36,31)(16,30,25,21)(17,20,26,29)(18,28,27,19)(37,42,96,89)(38,88,85,41)(39,40,86,87)(43,48,90,95)(44,94,91,47)(45,46,92,93)(49,80,75,54)(50,53,76,79)(51,78,77,52)(55,74,81,60)(56,59,82,73)(57,84,83,58)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,49)(2,82)(3,51)(4,84)(5,53)(6,74)(7,55)(8,76)(9,57)(10,78)(11,59)(12,80)(13,40)(14,94)(15,42)(16,96)(17,44)(18,86)(19,46)(20,88)(21,48)(22,90)(23,38)(24,92)(25,37)(26,91)(27,39)(28,93)(29,41)(30,95)(31,43)(32,85)(33,45)(34,87)(35,47)(36,89)(50,70)(52,72)(54,62)(56,64)(58,66)(60,68)(61,73)(63,75)(65,77)(67,79)(69,81)(71,83), (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), (1,69)(2,70)(3,71)(4,72)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,85)(45,86)(46,87)(47,88)(48,89)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,6,63,68)(2,67,64,5)(3,4,65,66)(7,12,69,62)(8,61,70,11)(9,10,71,72)(13,24,34,33)(14,32,35,23)(15,22,36,31)(16,30,25,21)(17,20,26,29)(18,28,27,19)(37,42,96,89)(38,88,85,41)(39,40,86,87)(43,48,90,95)(44,94,91,47)(45,46,92,93)(49,80,75,54)(50,53,76,79)(51,78,77,52)(55,74,81,60)(56,59,82,73)(57,84,83,58) );
G=PermutationGroup([(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,84),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(61,94),(62,95),(63,96),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93)], [(1,49),(2,82),(3,51),(4,84),(5,53),(6,74),(7,55),(8,76),(9,57),(10,78),(11,59),(12,80),(13,40),(14,94),(15,42),(16,96),(17,44),(18,86),(19,46),(20,88),(21,48),(22,90),(23,38),(24,92),(25,37),(26,91),(27,39),(28,93),(29,41),(30,95),(31,43),(32,85),(33,45),(34,87),(35,47),(36,89),(50,70),(52,72),(54,62),(56,64),(58,66),(60,68),(61,73),(63,75),(65,77),(67,79),(69,81),(71,83)], [(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)], [(1,69),(2,70),(3,71),(4,72),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,25),(23,26),(24,27),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,85),(45,86),(46,87),(47,88),(48,89),(49,81),(50,82),(51,83),(52,84),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,6,63,68),(2,67,64,5),(3,4,65,66),(7,12,69,62),(8,61,70,11),(9,10,71,72),(13,24,34,33),(14,32,35,23),(15,22,36,31),(16,30,25,21),(17,20,26,29),(18,28,27,19),(37,42,96,89),(38,88,85,41),(39,40,86,87),(43,48,90,95),(44,94,91,47),(45,46,92,93),(49,80,75,54),(50,53,76,79),(51,78,77,52),(55,74,81,60),(56,59,82,73),(57,84,83,58)])
Matrix representation ►G ⊆ GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 5 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,11,8],[12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,1,0,0,0,0,0,2,5] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 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 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C2×C23.11D6 | C23.11D6 | C2×C4×Dic3 | C2×D6⋊C4 | C2×C6.D4 | C6×C22⋊C4 | C22×Dic6 | C22×C3⋊D4 | C2×C22⋊C4 | C2×Dic3 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C22 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 8 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{11}D_6 % in TeX
G:=Group("C2xC2^3.11D6"); // GroupNames label
G:=SmallGroup(192,1050);
// by ID
G=gap.SmallGroup(192,1050);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,1571,297,6278]);
// 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,f*b*f^-1=b*c=c*b,e*b*e^-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=d*e^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀