direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C24⋊C22, C14.1702+ (1+4), C24⋊5(C2×C14), C22≀C2⋊8C14, C42⋊12(C2×C14), (C4×C28)⋊46C22, C4.4D4⋊15C14, (C23×C14)⋊5C22, (Q8×C14)⋊31C22, (C2×C14).381C24, (C2×C28).682C23, (D4×C14).223C22, C23.24(C22×C14), C22.55(C23×C14), C2.22(C7×2+ (1+4)), (C22×C14).107C23, (C2×Q8)⋊6(C2×C14), C22⋊C4⋊8(C2×C14), (C7×C22≀C2)⋊18C2, (C2×D4).36(C2×C14), (C7×C4.4D4)⋊35C2, (C7×C22⋊C4)⋊43C22, (C2×C4).41(C22×C14), SmallGroup(448,1344)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 506 in 260 conjugacy classes, 142 normal (6 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×26], C7, C2×C4 [×9], D4 [×9], Q8 [×3], C23 [×6], C23 [×6], C14 [×3], C14 [×6], C42 [×3], C22⋊C4 [×18], C2×D4 [×9], C2×Q8 [×3], C24 [×2], C28 [×9], C2×C14, C2×C14 [×26], C22≀C2 [×6], C4.4D4 [×9], C2×C28 [×9], C7×D4 [×9], C7×Q8 [×3], C22×C14 [×6], C22×C14 [×6], C24⋊C22, C4×C28 [×3], C7×C22⋊C4 [×18], D4×C14 [×9], Q8×C14 [×3], C23×C14 [×2], C7×C22≀C2 [×6], C7×C4.4D4 [×9], C7×C24⋊C22
Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C24, C2×C14 [×35], 2+ (1+4) [×3], C22×C14 [×15], C24⋊C22, C23×C14, C7×2+ (1+4) [×3], C7×C24⋊C22
Generators and relations
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fbf=be=eb, gbg=bde, gcg=cd=dc, ce=ec, fcf=cde, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >
►On 112 points
Generators in S112
(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)
(1 102)(2 103)(3 104)(4 105)(5 99)(6 100)(7 101)(8 77)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 70)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 83)(37 84)(38 78)(39 79)(40 80)(41 81)(42 82)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(57 111)(58 112)(59 106)(60 107)(61 108)(62 109)(63 110)
(1 74)(2 75)(3 76)(4 77)(5 71)(6 72)(7 73)(8 105)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 98)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 83)(23 84)(24 78)(25 79)(26 80)(27 81)(28 82)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 55)(37 56)(38 50)(39 51)(40 52)(41 53)(42 54)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(85 111)(86 112)(87 106)(88 107)(89 108)(90 109)(91 110)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
G:=sub<Sym(112)| (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), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,77)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,70)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(57,111)(58,112)(59,106)(60,107)(61,108)(62,109)(63,110), (1,74)(2,75)(3,76)(4,77)(5,71)(6,72)(7,73)(8,105)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,98)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,55)(37,56)(38,50)(39,51)(40,52)(41,53)(42,54)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(85,111)(86,112)(87,106)(88,107)(89,108)(90,109)(91,110), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)>;
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)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,77)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,70)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(57,111)(58,112)(59,106)(60,107)(61,108)(62,109)(63,110), (1,74)(2,75)(3,76)(4,77)(5,71)(6,72)(7,73)(8,105)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,98)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,55)(37,56)(38,50)(39,51)(40,52)(41,53)(42,54)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(85,111)(86,112)(87,106)(88,107)(89,108)(90,109)(91,110), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99) );
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),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(1,102),(2,103),(3,104),(4,105),(5,99),(6,100),(7,101),(8,77),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,70),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,55),(23,56),(24,50),(25,51),(26,52),(27,53),(28,54),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,83),(37,84),(38,78),(39,79),(40,80),(41,81),(42,82),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(57,111),(58,112),(59,106),(60,107),(61,108),(62,109),(63,110)], [(1,74),(2,75),(3,76),(4,77),(5,71),(6,72),(7,73),(8,105),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,98),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,83),(23,84),(24,78),(25,79),(26,80),(27,81),(28,82),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,55),(37,56),(38,50),(39,51),(40,52),(41,53),(42,54),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(85,111),(86,112),(87,106),(88,107),(89,108),(90,109),(91,110)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(50,67),(51,68),(52,69),(53,70),(54,64),(55,65),(56,66),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,95),(79,96),(80,97),(81,98),(82,92),(83,93),(84,94),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 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 | 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 |
| 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(8,GF(29))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[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],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28] >;
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4I | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 28A | ··· | 28BB |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | 2+ (1+4) | C7×2+ (1+4) |
| kernel | C7×C24⋊C22 | C7×C22≀C2 | C7×C4.4D4 | C24⋊C22 | C22≀C2 | C4.4D4 | C14 | C2 |
| # reps | 1 | 6 | 9 | 6 | 36 | 54 | 3 | 18 |
In GAP, Magma, Sage, TeX
C_7\times C_2^4\rtimes C_2^2
% in TeX
G:=Group("C7xC2^4:C2^2"); // GroupNames label
G:=SmallGroup(448,1344);
// by ID
G=gap.SmallGroup(448,1344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790,3579,9635,1690]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=f^2=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*b*f=b*e=e*b,g*b*g=b*d*e,g*c*g=c*d=d*c,c*e=e*c,f*c*f=c*d*e,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀