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, C22.37(C23×C14), (C23×C14).18C22, 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), (C22×C4).63(C2×C14), (C2×C4).30(C22×C14), (C7×C22⋊C4).151C22, SmallGroup(448,1326)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23⋊2Q8
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 >
Subgroups: 386 in 242 conjugacy classes, 162 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22⋊Q8, C2×C28, C2×C28, C7×Q8, C22×C14, C22×C14, C23⋊2Q8, C7×C22⋊C4, C7×C4⋊C4, C22×C28, Q8×C14, C23×C14, C14×C22⋊C4, C7×C22⋊Q8, C7×C23⋊2Q8
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2×Q8, C24, C2×C14, C22×Q8, 2+ 1+4, C7×Q8, C22×C14, C23⋊2Q8, Q8×C14, C23×C14, Q8×C2×C14, C7×2+ 1+4, C7×C23⋊2Q8
►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 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)(50 63)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(64 75)(65 76)(66 77)(67 71)(68 72)(69 73)(70 74)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 103)(93 104)(94 105)(95 99)(96 100)(97 101)(98 102)
(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 106)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)
(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 106)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)
(1 63 35 74)(2 57 29 75)(3 58 30 76)(4 59 31 77)(5 60 32 71)(6 61 33 72)(7 62 34 73)(8 85 18 103)(9 86 19 104)(10 87 20 105)(11 88 21 99)(12 89 15 100)(13 90 16 101)(14 91 17 102)(22 83 111 96)(23 84 112 97)(24 78 106 98)(25 79 107 92)(26 80 108 93)(27 81 109 94)(28 82 110 95)(36 51 48 64)(37 52 49 65)(38 53 43 66)(39 54 44 67)(40 55 45 68)(41 56 46 69)(42 50 47 70)
(1 91 35 102)(2 85 29 103)(3 86 30 104)(4 87 31 105)(5 88 32 99)(6 89 33 100)(7 90 34 101)(8 75 18 57)(9 76 19 58)(10 77 20 59)(11 71 21 60)(12 72 15 61)(13 73 16 62)(14 74 17 63)(22 68 111 55)(23 69 112 56)(24 70 106 50)(25 64 107 51)(26 65 108 52)(27 66 109 53)(28 67 110 54)(36 79 48 92)(37 80 49 93)(38 81 43 94)(39 82 44 95)(40 83 45 96)(41 84 46 97)(42 78 47 98)
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)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47)(50,63)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(64,75)(65,76)(66,77)(67,71)(68,72)(69,73)(70,74)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,103)(93,104)(94,105)(95,99)(96,100)(97,101)(98,102), (8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,106)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,106)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (1,63,35,74)(2,57,29,75)(3,58,30,76)(4,59,31,77)(5,60,32,71)(6,61,33,72)(7,62,34,73)(8,85,18,103)(9,86,19,104)(10,87,20,105)(11,88,21,99)(12,89,15,100)(13,90,16,101)(14,91,17,102)(22,83,111,96)(23,84,112,97)(24,78,106,98)(25,79,107,92)(26,80,108,93)(27,81,109,94)(28,82,110,95)(36,51,48,64)(37,52,49,65)(38,53,43,66)(39,54,44,67)(40,55,45,68)(41,56,46,69)(42,50,47,70), (1,91,35,102)(2,85,29,103)(3,86,30,104)(4,87,31,105)(5,88,32,99)(6,89,33,100)(7,90,34,101)(8,75,18,57)(9,76,19,58)(10,77,20,59)(11,71,21,60)(12,72,15,61)(13,73,16,62)(14,74,17,63)(22,68,111,55)(23,69,112,56)(24,70,106,50)(25,64,107,51)(26,65,108,52)(27,66,109,53)(28,67,110,54)(36,79,48,92)(37,80,49,93)(38,81,43,94)(39,82,44,95)(40,83,45,96)(41,84,46,97)(42,78,47,98)>;
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)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47)(50,63)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62)(64,75)(65,76)(66,77)(67,71)(68,72)(69,73)(70,74)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,103)(93,104)(94,105)(95,99)(96,100)(97,101)(98,102), (8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,106)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,106)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (1,63,35,74)(2,57,29,75)(3,58,30,76)(4,59,31,77)(5,60,32,71)(6,61,33,72)(7,62,34,73)(8,85,18,103)(9,86,19,104)(10,87,20,105)(11,88,21,99)(12,89,15,100)(13,90,16,101)(14,91,17,102)(22,83,111,96)(23,84,112,97)(24,78,106,98)(25,79,107,92)(26,80,108,93)(27,81,109,94)(28,82,110,95)(36,51,48,64)(37,52,49,65)(38,53,43,66)(39,54,44,67)(40,55,45,68)(41,56,46,69)(42,50,47,70), (1,91,35,102)(2,85,29,103)(3,86,30,104)(4,87,31,105)(5,88,32,99)(6,89,33,100)(7,90,34,101)(8,75,18,57)(9,76,19,58)(10,77,20,59)(11,71,21,60)(12,72,15,61)(13,73,16,62)(14,74,17,63)(22,68,111,55)(23,69,112,56)(24,70,106,50)(25,64,107,51)(26,65,108,52)(27,66,109,53)(28,67,110,54)(36,79,48,92)(37,80,49,93)(38,81,43,94)(39,82,44,95)(40,83,45,96)(41,84,46,97)(42,78,47,98) );
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),(36,48),(37,49),(38,43),(39,44),(40,45),(41,46),(42,47),(50,63),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62),(64,75),(65,76),(66,77),(67,71),(68,72),(69,73),(70,74),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,103),(93,104),(94,105),(95,99),(96,100),(97,101),(98,102)], [(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,106),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98)], [(1,47),(2,48),(3,49),(4,43),(5,44),(6,45),(7,46),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,106),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98)], [(1,63,35,74),(2,57,29,75),(3,58,30,76),(4,59,31,77),(5,60,32,71),(6,61,33,72),(7,62,34,73),(8,85,18,103),(9,86,19,104),(10,87,20,105),(11,88,21,99),(12,89,15,100),(13,90,16,101),(14,91,17,102),(22,83,111,96),(23,84,112,97),(24,78,106,98),(25,79,107,92),(26,80,108,93),(27,81,109,94),(28,82,110,95),(36,51,48,64),(37,52,49,65),(38,53,43,66),(39,54,44,67),(40,55,45,68),(41,56,46,69),(42,50,47,70)], [(1,91,35,102),(2,85,29,103),(3,86,30,104),(4,87,31,105),(5,88,32,99),(6,89,33,100),(7,90,34,101),(8,75,18,57),(9,76,19,58),(10,77,20,59),(11,71,21,60),(12,72,15,61),(13,73,16,62),(14,74,17,63),(22,68,111,55),(23,69,112,56),(24,70,106,50),(25,64,107,51),(26,65,108,52),(27,66,109,53),(28,67,110,54),(36,79,48,92),(37,80,49,93),(38,81,43,94),(39,82,44,95),(40,83,45,96),(41,84,46,97),(42,78,47,98)]])
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 |
Matrix representation of C7×C23⋊2Q8 ►in 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] >;
C7×C23⋊2Q8 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