direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23⋊2Q8, C14.1582+ (1+4), C23⋊3(C7×Q8), (C22×C14)⋊2Q8, C22⋊Q8⋊10C14, C22.4(Q8×C14), C24.21(C2×C14), (Q8×C14)⋊29C22, C14.61(C22×Q8), (C2×C14).363C24, (C2×C28).672C23, (C23×C14).18C22, C22.37(C23×C14), C23.39(C22×C14), C2.10(C7×2+ (1+4)), (C22×C28).451C22, (C22×C14).262C23, C4⋊C4⋊4(C2×C14), C2.7(Q8×C2×C14), (C2×Q8)⋊4(C2×C14), (C7×C4⋊C4)⋊38C22, (C7×C22⋊Q8)⋊37C2, (C2×C14).17(C2×Q8), (C2×C22⋊C4).13C14, (C14×C22⋊C4).33C2, C22⋊C4.17(C2×C14), (C2×C4).30(C22×C14), (C22×C4).63(C2×C14), (C7×C22⋊C4).151C22, SmallGroup(448,1326)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 386 in 242 conjugacy classes, 162 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×10], C7, C2×C4 [×12], C2×C4 [×6], Q8 [×4], C23 [×7], C23 [×2], C14, C14 [×2], C14 [×6], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C2×Q8 [×4], C24, C28 [×12], C2×C14, C2×C14 [×6], C2×C14 [×10], C2×C22⋊C4 [×3], C22⋊Q8 [×12], C2×C28 [×12], C2×C28 [×6], C7×Q8 [×4], C22×C14 [×7], C22×C14 [×2], C23⋊2Q8, C7×C22⋊C4 [×12], C7×C4⋊C4 [×12], C22×C28 [×6], Q8×C14 [×4], C23×C14, C14×C22⋊C4 [×3], C7×C22⋊Q8 [×12], C7×C23⋊2Q8
Quotients:
C1, C2 [×15], C22 [×35], C7, Q8 [×4], C23 [×15], C14 [×15], C2×Q8 [×6], C24, C2×C14 [×35], C22×Q8, 2+ (1+4) [×2], C7×Q8 [×4], C22×C14 [×15], C23⋊2Q8, Q8×C14 [×6], C23×C14, Q8×C2×C14, C7×2+ (1+4) [×2], C7×C23⋊2Q8
Generators and relations
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >
►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 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 43)(50 63)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 101)(93 102)(94 103)(95 104)(96 105)(97 99)(98 100)
(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)
(1 63 35 66)(2 57 29 67)(3 58 30 68)(4 59 31 69)(5 60 32 70)(6 61 33 64)(7 62 34 65)(8 93 18 90)(9 94 19 91)(10 95 20 85)(11 96 21 86)(12 97 15 87)(13 98 16 88)(14 92 17 89)(22 83 111 101)(23 84 112 102)(24 78 106 103)(25 79 107 104)(26 80 108 105)(27 81 109 99)(28 82 110 100)(36 72 44 54)(37 73 45 55)(38 74 46 56)(39 75 47 50)(40 76 48 51)(41 77 49 52)(42 71 43 53)
(1 91 35 94)(2 85 29 95)(3 86 30 96)(4 87 31 97)(5 88 32 98)(6 89 33 92)(7 90 34 93)(8 62 18 65)(9 63 19 66)(10 57 20 67)(11 58 21 68)(12 59 15 69)(13 60 16 70)(14 61 17 64)(22 73 111 55)(23 74 112 56)(24 75 106 50)(25 76 107 51)(26 77 108 52)(27 71 109 53)(28 72 110 54)(36 100 44 82)(37 101 45 83)(38 102 46 84)(39 103 47 78)(40 104 48 79)(41 105 49 80)(42 99 43 81)
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), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,63)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,63,35,66)(2,57,29,67)(3,58,30,68)(4,59,31,69)(5,60,32,70)(6,61,33,64)(7,62,34,65)(8,93,18,90)(9,94,19,91)(10,95,20,85)(11,96,21,86)(12,97,15,87)(13,98,16,88)(14,92,17,89)(22,83,111,101)(23,84,112,102)(24,78,106,103)(25,79,107,104)(26,80,108,105)(27,81,109,99)(28,82,110,100)(36,72,44,54)(37,73,45,55)(38,74,46,56)(39,75,47,50)(40,76,48,51)(41,77,49,52)(42,71,43,53), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,62,18,65)(9,63,19,66)(10,57,20,67)(11,58,21,68)(12,59,15,69)(13,60,16,70)(14,61,17,64)(22,73,111,55)(23,74,112,56)(24,75,106,50)(25,76,107,51)(26,77,108,52)(27,71,109,53)(28,72,110,54)(36,100,44,82)(37,101,45,83)(38,102,46,84)(39,103,47,78)(40,104,48,79)(41,105,49,80)(42,99,43,81)>;
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), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,63)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,63,35,66)(2,57,29,67)(3,58,30,68)(4,59,31,69)(5,60,32,70)(6,61,33,64)(7,62,34,65)(8,93,18,90)(9,94,19,91)(10,95,20,85)(11,96,21,86)(12,97,15,87)(13,98,16,88)(14,92,17,89)(22,83,111,101)(23,84,112,102)(24,78,106,103)(25,79,107,104)(26,80,108,105)(27,81,109,99)(28,82,110,100)(36,72,44,54)(37,73,45,55)(38,74,46,56)(39,75,47,50)(40,76,48,51)(41,77,49,52)(42,71,43,53), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,62,18,65)(9,63,19,66)(10,57,20,67)(11,58,21,68)(12,59,15,69)(13,60,16,70)(14,61,17,64)(22,73,111,55)(23,74,112,56)(24,75,106,50)(25,76,107,51)(26,77,108,52)(27,71,109,53)(28,72,110,54)(36,100,44,82)(37,101,45,83)(38,102,46,84)(39,103,47,78)(40,104,48,79)(41,105,49,80)(42,99,43,81) );
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)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(36,44),(37,45),(38,46),(39,47),(40,48),(41,49),(42,43),(50,63),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62),(64,73),(65,74),(66,75),(67,76),(68,77),(69,71),(70,72),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,101),(93,102),(94,103),(95,104),(96,105),(97,99),(98,100)], [(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103)], [(1,63,35,66),(2,57,29,67),(3,58,30,68),(4,59,31,69),(5,60,32,70),(6,61,33,64),(7,62,34,65),(8,93,18,90),(9,94,19,91),(10,95,20,85),(11,96,21,86),(12,97,15,87),(13,98,16,88),(14,92,17,89),(22,83,111,101),(23,84,112,102),(24,78,106,103),(25,79,107,104),(26,80,108,105),(27,81,109,99),(28,82,110,100),(36,72,44,54),(37,73,45,55),(38,74,46,56),(39,75,47,50),(40,76,48,51),(41,77,49,52),(42,71,43,53)], [(1,91,35,94),(2,85,29,95),(3,86,30,96),(4,87,31,97),(5,88,32,98),(6,89,33,92),(7,90,34,93),(8,62,18,65),(9,63,19,66),(10,57,20,67),(11,58,21,68),(12,59,15,69),(13,60,16,70),(14,61,17,64),(22,73,111,55),(23,74,112,56),(24,75,106,50),(25,76,107,51),(26,77,108,52),(27,71,109,53),(28,72,110,54),(36,100,44,82),(37,101,45,83),(38,102,46,84),(39,103,47,78),(40,104,48,79),(41,105,49,80),(42,99,43,81)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 0 | 20 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 2 | 0 | 28 | 0 |
| 0 | 0 | 0 | 2 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 13 | 28 | 0 |
| 0 | 0 | 13 | 27 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 1 |
| 0 | 0 | 27 | 0 | 1 | 0 |
| 27 | 16 | 0 | 0 | 0 | 0 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 16 | 2 | 0 |
| 0 | 0 | 16 | 2 | 0 | 2 |
| 0 | 0 | 1 | 0 | 2 | 13 |
| 0 | 0 | 0 | 1 | 13 | 27 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,2,0,0,0,0,28,0,2,0,0,0,0,28,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,2,13,0,0,0,1,13,27,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,27,0,0,1,0,2,0,0,0,0,0,0,1,0,0,0,0,1,0],[27,16,0,0,0,0,16,2,0,0,0,0,0,0,27,16,1,0,0,0,16,2,0,1,0,0,2,0,2,13,0,0,0,2,13,27] >;
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 28A | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | Q8 | C7×Q8 | 2+ (1+4) | C7×2+ (1+4) |
| kernel | C7×C23⋊2Q8 | C14×C22⋊C4 | C7×C22⋊Q8 | C23⋊2Q8 | C2×C22⋊C4 | C22⋊Q8 | C22×C14 | C23 | C14 | C2 |
| # reps | 1 | 3 | 12 | 6 | 18 | 72 | 4 | 24 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_7\times C_2^3\rtimes_2Q_8
% in TeX
G:=Group("C7xC2^3:2Q8"); // GroupNames label
G:=SmallGroup(448,1326);
// by ID
G=gap.SmallGroup(448,1326);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,4790,1227,1192,3363]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀