metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.34D14, C14.282+ (1+4), (C2×D4)⋊6D14, C22≀C2⋊5D7, C22⋊C4⋊7D14, (C2×Dic7)⋊8D4, C23⋊D14⋊6C2, C28⋊D4⋊12C2, (D4×C14)⋊9C22, C24⋊D7⋊7C2, Dic7.4(C2×D4), C22⋊D28⋊10C2, D14⋊D4⋊14C2, C22.41(D4×D7), D14⋊C4⋊13C22, Dic7⋊D4⋊4C2, (C2×D28)⋊20C22, (C2×C28).30C23, C14.58(C22×D4), (C2×C14).136C24, Dic7⋊C4⋊11C22, C7⋊2(C22.29C24), (C4×Dic7)⋊16C22, C23.D7⋊16C22, C2.30(D4⋊6D14), Dic7.D4⋊13C2, (C2×Dic14)⋊21C22, C23.11D14⋊3C2, (C22×C14).10C23, (C23×C14).69C22, (C2×Dic7).61C23, (C23×D7).44C22, (C22×D7).55C23, C22.157(C23×D7), C23.109(C22×D7), (C22×Dic7)⋊15C22, C2.31(C2×D4×D7), (C2×C4×D7)⋊9C22, (C7×C22≀C2)⋊7C2, (C2×D4⋊2D7)⋊7C2, (C2×C14).55(C2×D4), (C2×C7⋊D4)⋊9C22, (C22×C7⋊D4)⋊10C2, (C7×C22⋊C4)⋊7C22, (C2×C4).30(C22×D7), SmallGroup(448,1045)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1804 in 334 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×10], C22, C22 [×2], C22 [×28], C7, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×2], C23 [×2], C23 [×11], D7 [×3], C14, C14 [×2], C14 [×5], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24, C24, Dic7 [×4], Dic7 [×3], C28 [×3], D14 [×13], C2×C14, C2×C14 [×2], C2×C14 [×15], C42⋊C2, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×2], D28 [×2], C2×Dic7 [×3], C2×Dic7 [×6], C2×Dic7 [×2], C7⋊D4 [×16], C2×C28, C2×C28 [×2], C7×D4 [×4], C22×D7, C22×D7 [×2], C22×D7 [×4], C22×C14 [×2], C22×C14 [×2], C22×C14 [×4], C22.29C24, C4×Dic7 [×2], Dic7⋊C4 [×2], D14⋊C4 [×4], C23.D7, C23.D7 [×2], C7×C22⋊C4, C7×C22⋊C4 [×2], C2×Dic14, C2×C4×D7, C2×D28 [×2], D4⋊2D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×2], C2×C7⋊D4 [×8], C2×C7⋊D4 [×4], D4×C14, D4×C14 [×2], C23×D7, C23×C14, C23.11D14, C22⋊D28, D14⋊D4 [×2], Dic7.D4 [×2], C23⋊D14, Dic7⋊D4 [×2], C28⋊D4 [×2], C24⋊D7, C7×C22≀C2, C2×D4⋊2D7, C22×C7⋊D4, C24.34D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4) [×2], C22×D7 [×7], C22.29C24, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14 [×2], C24.34D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >
►On 112 points
Generators in S112
(1 94)(2 18)(3 96)(4 20)(5 98)(6 22)(7 86)(8 24)(9 88)(10 26)(11 90)(12 28)(13 92)(14 16)(15 71)(17 73)(19 75)(21 77)(23 79)(25 81)(27 83)(29 44)(30 59)(31 46)(32 61)(33 48)(34 63)(35 50)(36 65)(37 52)(38 67)(39 54)(40 69)(41 56)(42 57)(43 111)(45 99)(47 101)(49 103)(51 105)(53 107)(55 109)(58 112)(60 100)(62 102)(64 104)(66 106)(68 108)(70 110)(72 93)(74 95)(76 97)(78 85)(80 87)(82 89)(84 91)
(1 48)(2 34)(3 50)(4 36)(5 52)(6 38)(7 54)(8 40)(9 56)(10 42)(11 44)(12 30)(13 46)(14 32)(15 100)(16 61)(17 102)(18 63)(19 104)(20 65)(21 106)(22 67)(23 108)(24 69)(25 110)(26 57)(27 112)(28 59)(29 90)(31 92)(33 94)(35 96)(37 98)(39 86)(41 88)(43 89)(45 91)(47 93)(49 95)(51 97)(53 85)(55 87)(58 83)(60 71)(62 73)(64 75)(66 77)(68 79)(70 81)(72 101)(74 103)(76 105)(78 107)(80 109)(82 111)(84 99)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 71)(14 72)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 112)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 43)(57 111)(58 112)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)
(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 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 14 94 93)(2 92 95 13)(3 12 96 91)(4 90 97 11)(5 10 98 89)(6 88 85 9)(7 8 86 87)(15 18 71 74)(16 73 72 17)(19 28 75 84)(20 83 76 27)(21 26 77 82)(22 81 78 25)(23 24 79 80)(29 105 44 65)(30 64 45 104)(31 103 46 63)(32 62 47 102)(33 101 48 61)(34 60 49 100)(35 99 50 59)(36 58 51 112)(37 111 52 57)(38 70 53 110)(39 109 54 69)(40 68 55 108)(41 107 56 67)(42 66 43 106)
G:=sub<Sym(112)| (1,94)(2,18)(3,96)(4,20)(5,98)(6,22)(7,86)(8,24)(9,88)(10,26)(11,90)(12,28)(13,92)(14,16)(15,71)(17,73)(19,75)(21,77)(23,79)(25,81)(27,83)(29,44)(30,59)(31,46)(32,61)(33,48)(34,63)(35,50)(36,65)(37,52)(38,67)(39,54)(40,69)(41,56)(42,57)(43,111)(45,99)(47,101)(49,103)(51,105)(53,107)(55,109)(58,112)(60,100)(62,102)(64,104)(66,106)(68,108)(70,110)(72,93)(74,95)(76,97)(78,85)(80,87)(82,89)(84,91), (1,48)(2,34)(3,50)(4,36)(5,52)(6,38)(7,54)(8,40)(9,56)(10,42)(11,44)(12,30)(13,46)(14,32)(15,100)(16,61)(17,102)(18,63)(19,104)(20,65)(21,106)(22,67)(23,108)(24,69)(25,110)(26,57)(27,112)(28,59)(29,90)(31,92)(33,94)(35,96)(37,98)(39,86)(41,88)(43,89)(45,91)(47,93)(49,95)(51,97)(53,85)(55,87)(58,83)(60,71)(62,73)(64,75)(66,77)(68,79)(70,81)(72,101)(74,103)(76,105)(78,107)(80,109)(82,111)(84,99), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,71)(14,72)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,112)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (1,94)(2,95)(3,96)(4,97)(5,98)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,43)(57,111)(58,112)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,14,94,93)(2,92,95,13)(3,12,96,91)(4,90,97,11)(5,10,98,89)(6,88,85,9)(7,8,86,87)(15,18,71,74)(16,73,72,17)(19,28,75,84)(20,83,76,27)(21,26,77,82)(22,81,78,25)(23,24,79,80)(29,105,44,65)(30,64,45,104)(31,103,46,63)(32,62,47,102)(33,101,48,61)(34,60,49,100)(35,99,50,59)(36,58,51,112)(37,111,52,57)(38,70,53,110)(39,109,54,69)(40,68,55,108)(41,107,56,67)(42,66,43,106)>;
G:=Group( (1,94)(2,18)(3,96)(4,20)(5,98)(6,22)(7,86)(8,24)(9,88)(10,26)(11,90)(12,28)(13,92)(14,16)(15,71)(17,73)(19,75)(21,77)(23,79)(25,81)(27,83)(29,44)(30,59)(31,46)(32,61)(33,48)(34,63)(35,50)(36,65)(37,52)(38,67)(39,54)(40,69)(41,56)(42,57)(43,111)(45,99)(47,101)(49,103)(51,105)(53,107)(55,109)(58,112)(60,100)(62,102)(64,104)(66,106)(68,108)(70,110)(72,93)(74,95)(76,97)(78,85)(80,87)(82,89)(84,91), (1,48)(2,34)(3,50)(4,36)(5,52)(6,38)(7,54)(8,40)(9,56)(10,42)(11,44)(12,30)(13,46)(14,32)(15,100)(16,61)(17,102)(18,63)(19,104)(20,65)(21,106)(22,67)(23,108)(24,69)(25,110)(26,57)(27,112)(28,59)(29,90)(31,92)(33,94)(35,96)(37,98)(39,86)(41,88)(43,89)(45,91)(47,93)(49,95)(51,97)(53,85)(55,87)(58,83)(60,71)(62,73)(64,75)(66,77)(68,79)(70,81)(72,101)(74,103)(76,105)(78,107)(80,109)(82,111)(84,99), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,71)(14,72)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,112)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (1,94)(2,95)(3,96)(4,97)(5,98)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,43)(57,111)(58,112)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,14,94,93)(2,92,95,13)(3,12,96,91)(4,90,97,11)(5,10,98,89)(6,88,85,9)(7,8,86,87)(15,18,71,74)(16,73,72,17)(19,28,75,84)(20,83,76,27)(21,26,77,82)(22,81,78,25)(23,24,79,80)(29,105,44,65)(30,64,45,104)(31,103,46,63)(32,62,47,102)(33,101,48,61)(34,60,49,100)(35,99,50,59)(36,58,51,112)(37,111,52,57)(38,70,53,110)(39,109,54,69)(40,68,55,108)(41,107,56,67)(42,66,43,106) );
G=PermutationGroup([(1,94),(2,18),(3,96),(4,20),(5,98),(6,22),(7,86),(8,24),(9,88),(10,26),(11,90),(12,28),(13,92),(14,16),(15,71),(17,73),(19,75),(21,77),(23,79),(25,81),(27,83),(29,44),(30,59),(31,46),(32,61),(33,48),(34,63),(35,50),(36,65),(37,52),(38,67),(39,54),(40,69),(41,56),(42,57),(43,111),(45,99),(47,101),(49,103),(51,105),(53,107),(55,109),(58,112),(60,100),(62,102),(64,104),(66,106),(68,108),(70,110),(72,93),(74,95),(76,97),(78,85),(80,87),(82,89),(84,91)], [(1,48),(2,34),(3,50),(4,36),(5,52),(6,38),(7,54),(8,40),(9,56),(10,42),(11,44),(12,30),(13,46),(14,32),(15,100),(16,61),(17,102),(18,63),(19,104),(20,65),(21,106),(22,67),(23,108),(24,69),(25,110),(26,57),(27,112),(28,59),(29,90),(31,92),(33,94),(35,96),(37,98),(39,86),(41,88),(43,89),(45,91),(47,93),(49,95),(51,97),(53,85),(55,87),(58,83),(60,71),(62,73),(64,75),(66,77),(68,79),(70,81),(72,101),(74,103),(76,105),(78,107),(80,109),(82,111),(84,99)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,71),(14,72),(15,92),(16,93),(17,94),(18,95),(19,96),(20,97),(21,98),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,112),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,92),(14,93),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,43),(57,111),(58,112),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,14,94,93),(2,92,95,13),(3,12,96,91),(4,90,97,11),(5,10,98,89),(6,88,85,9),(7,8,86,87),(15,18,71,74),(16,73,72,17),(19,28,75,84),(20,83,76,27),(21,26,77,82),(22,81,78,25),(23,24,79,80),(29,105,44,65),(30,64,45,104),(31,103,46,63),(32,62,47,102),(33,101,48,61),(34,60,49,100),(35,99,50,59),(36,58,51,112),(37,111,52,57),(38,70,53,110),(39,109,54,69),(40,68,55,108),(41,107,56,67),(42,66,43,106)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 17 | 1 | 0 |
| 0 | 0 | 0 | 0 | 24 | 18 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 9 | 24 | 13 |
| 0 | 0 | 0 | 0 | 24 | 1 | 16 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 11 | 18 | 12 | 17 | 0 | 0 | 0 | 0 |
| 11 | 21 | 12 | 15 | 0 | 0 | 0 | 0 |
| 17 | 12 | 18 | 11 | 0 | 0 | 0 | 0 |
| 17 | 14 | 18 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 | 6 | 27 |
| 0 | 0 | 0 | 0 | 26 | 1 | 7 | 8 |
| 0 | 0 | 0 | 0 | 3 | 4 | 15 | 28 |
| 0 | 0 | 0 | 0 | 20 | 5 | 10 | 10 |
| 18 | 11 | 17 | 12 | 0 | 0 | 0 | 0 |
| 21 | 11 | 15 | 12 | 0 | 0 | 0 | 0 |
| 12 | 17 | 11 | 18 | 0 | 0 | 0 | 0 |
| 14 | 17 | 8 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 21 | 2 | 23 |
| 0 | 0 | 0 | 0 | 25 | 14 | 21 | 22 |
| 0 | 0 | 0 | 0 | 4 | 20 | 1 | 14 |
| 0 | 0 | 0 | 0 | 11 | 14 | 19 | 19 |
G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,9,24,0,0,0,0,0,28,17,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,7,10,24,0,0,0,0,16,18,9,1,0,0,0,0,0,0,24,16,0,0,0,0,0,0,13,5],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[11,11,17,17,0,0,0,0,18,21,12,14,0,0,0,0,12,12,18,18,0,0,0,0,17,15,11,8,0,0,0,0,0,0,0,0,3,26,3,20,0,0,0,0,4,1,4,5,0,0,0,0,6,7,15,10,0,0,0,0,27,8,28,10],[18,21,12,14,0,0,0,0,11,11,17,17,0,0,0,0,17,15,11,8,0,0,0,0,12,12,18,18,0,0,0,0,0,0,0,0,24,25,4,11,0,0,0,0,21,14,20,14,0,0,0,0,2,21,1,19,0,0,0,0,23,22,14,19] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 |
| kernel | C24.34D14 | C23.11D14 | C22⋊D28 | D14⋊D4 | Dic7.D4 | C23⋊D14 | Dic7⋊D4 | C28⋊D4 | C24⋊D7 | C7×C22≀C2 | C2×D4⋊2D7 | C22×C7⋊D4 | C2×Dic7 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 9 | 9 | 3 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_2^4._{34}D_{14} % in TeX
G:=Group("C2^4.34D14"); // GroupNames label
G:=SmallGroup(448,1045);
// by ID
G=gap.SmallGroup(448,1045);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
// generators/relations
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ℤ
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