metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊26D14, (C4×D7)⋊13D4, (C2×Q8)⋊15D14, C4.186(D4×D7), C22⋊Q8⋊26D7, D14⋊4(C4○D4), C28⋊1D4⋊23C2, C28⋊7D4⋊35C2, D14.42(C2×D4), C28.231(C2×D4), D28⋊C4⋊24C2, C22⋊D28⋊15C2, (Q8×C14)⋊6C22, D14⋊C4⋊19C22, D14⋊3Q8⋊13C2, (C2×D28)⋊24C22, C4⋊Dic7⋊35C22, C22⋊C4.56D14, Dic7.64(C2×D4), C14.73(C22×D4), Dic7⋊4D4⋊14C2, D14.5D4⋊15C2, (C2×C28).598C23, (C2×C14).171C24, Dic7⋊C4⋊17C22, C7⋊5(C22.19C24), C22⋊1(Q8⋊2D7), (C4×Dic7)⋊27C22, (C22×C4).372D14, (C2×Dic7).86C23, C22.192(C23×D7), C23.188(C22×D7), (C22×C28).251C22, (C22×C14).199C23, (C22×D7).193C23, (C23×D7).109C22, (C22×Dic7).225C22, C2.46(C2×D4×D7), (D7×C22×C4)⋊5C2, C2.48(D7×C4○D4), (C2×C4×D7)⋊17C22, C4⋊C4⋊7D7⋊24C2, (C7×C22⋊Q8)⋊7C2, (C2×C14)⋊6(C4○D4), (C7×C4⋊C4)⋊18C22, (C2×Q8⋊2D7)⋊5C2, C14.160(C2×C4○D4), C2.16(C2×Q8⋊2D7), (C2×C4).46(C22×D7), (C2×C7⋊D4).38C22, (C7×C22⋊C4).26C22, SmallGroup(448,1080)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1692 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], D4 [×14], Q8 [×2], C23, C23 [×10], D7 [×6], C14 [×3], C14 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×11], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic7 [×2], Dic7 [×3], C28 [×2], C28 [×5], D14 [×4], D14 [×18], C2×C14, C2×C14 [×2], C2×C14 [×2], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×2], C23×C4, C2×C4○D4, C4×D7 [×4], C4×D7 [×10], D28 [×10], C2×Dic7 [×2], C2×Dic7 [×2], C2×Dic7 [×2], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7 [×2], C22×D7 [×2], C22×D7 [×6], C22×C14, C22.19C24, C4×Dic7 [×2], Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4 [×8], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7 [×4], C2×C4×D7 [×2], C2×C4×D7 [×4], C2×D28, C2×D28 [×4], Q8⋊2D7 [×4], C22×Dic7, C2×C7⋊D4 [×2], C22×C28, Q8×C14, C23×D7, Dic7⋊4D4 [×2], C22⋊D28 [×2], C4⋊C4⋊7D7, D28⋊C4 [×2], D14.5D4 [×2], C28⋊1D4, C28⋊7D4, D14⋊3Q8, C7×C22⋊Q8, D7×C22×C4, C2×Q8⋊2D7, C4⋊C4⋊26D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C22×D7 [×7], C22.19C24, D4×D7 [×2], Q8⋊2D7 [×2], C23×D7, C2×D4×D7, C2×Q8⋊2D7, D7×C4○D4, C4⋊C4⋊26D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >
►On 112 points
(1 41 19 25)(2 42 20 26)(3 36 21 27)(4 37 15 28)(5 38 16 22)(6 39 17 23)(7 40 18 24)(8 47 33 51)(9 48 34 52)(10 49 35 53)(11 43 29 54)(12 44 30 55)(13 45 31 56)(14 46 32 50)(57 95 107 80)(58 96 108 81)(59 97 109 82)(60 98 110 83)(61 85 111 84)(62 86 112 71)(63 87 99 72)(64 88 100 73)(65 89 101 74)(66 90 102 75)(67 91 103 76)(68 92 104 77)(69 93 105 78)(70 94 106 79)
(1 74 12 96)(2 82 13 90)(3 76 14 98)(4 84 8 92)(5 78 9 86)(6 72 10 94)(7 80 11 88)(15 85 33 77)(16 93 34 71)(17 87 35 79)(18 95 29 73)(19 89 30 81)(20 97 31 75)(21 91 32 83)(22 69 52 112)(23 63 53 106)(24 57 54 100)(25 65 55 108)(26 59 56 102)(27 67 50 110)(28 61 51 104)(36 103 46 60)(37 111 47 68)(38 105 48 62)(39 99 49 70)(40 107 43 64)(41 101 44 58)(42 109 45 66)
(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 11)(2 10)(3 9)(4 8)(5 14)(6 13)(7 12)(15 33)(16 32)(17 31)(18 30)(19 29)(20 35)(21 34)(22 50)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(42 49)(57 58)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 92)(86 91)(87 90)(88 89)(93 98)(94 97)(95 96)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(112)| (1,41,19,25)(2,42,20,26)(3,36,21,27)(4,37,15,28)(5,38,16,22)(6,39,17,23)(7,40,18,24)(8,47,33,51)(9,48,34,52)(10,49,35,53)(11,43,29,54)(12,44,30,55)(13,45,31,56)(14,46,32,50)(57,95,107,80)(58,96,108,81)(59,97,109,82)(60,98,110,83)(61,85,111,84)(62,86,112,71)(63,87,99,72)(64,88,100,73)(65,89,101,74)(66,90,102,75)(67,91,103,76)(68,92,104,77)(69,93,105,78)(70,94,106,79), (1,74,12,96)(2,82,13,90)(3,76,14,98)(4,84,8,92)(5,78,9,86)(6,72,10,94)(7,80,11,88)(15,85,33,77)(16,93,34,71)(17,87,35,79)(18,95,29,73)(19,89,30,81)(20,97,31,75)(21,91,32,83)(22,69,52,112)(23,63,53,106)(24,57,54,100)(25,65,55,108)(26,59,56,102)(27,67,50,110)(28,61,51,104)(36,103,46,60)(37,111,47,68)(38,105,48,62)(39,99,49,70)(40,107,43,64)(41,101,44,58)(42,109,45,66), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,33)(16,32)(17,31)(18,30)(19,29)(20,35)(21,34)(22,50)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(42,49)(57,58)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,92)(86,91)(87,90)(88,89)(93,98)(94,97)(95,96)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,41,19,25)(2,42,20,26)(3,36,21,27)(4,37,15,28)(5,38,16,22)(6,39,17,23)(7,40,18,24)(8,47,33,51)(9,48,34,52)(10,49,35,53)(11,43,29,54)(12,44,30,55)(13,45,31,56)(14,46,32,50)(57,95,107,80)(58,96,108,81)(59,97,109,82)(60,98,110,83)(61,85,111,84)(62,86,112,71)(63,87,99,72)(64,88,100,73)(65,89,101,74)(66,90,102,75)(67,91,103,76)(68,92,104,77)(69,93,105,78)(70,94,106,79), (1,74,12,96)(2,82,13,90)(3,76,14,98)(4,84,8,92)(5,78,9,86)(6,72,10,94)(7,80,11,88)(15,85,33,77)(16,93,34,71)(17,87,35,79)(18,95,29,73)(19,89,30,81)(20,97,31,75)(21,91,32,83)(22,69,52,112)(23,63,53,106)(24,57,54,100)(25,65,55,108)(26,59,56,102)(27,67,50,110)(28,61,51,104)(36,103,46,60)(37,111,47,68)(38,105,48,62)(39,99,49,70)(40,107,43,64)(41,101,44,58)(42,109,45,66), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,33)(16,32)(17,31)(18,30)(19,29)(20,35)(21,34)(22,50)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(42,49)(57,58)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,92)(86,91)(87,90)(88,89)(93,98)(94,97)(95,96)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([(1,41,19,25),(2,42,20,26),(3,36,21,27),(4,37,15,28),(5,38,16,22),(6,39,17,23),(7,40,18,24),(8,47,33,51),(9,48,34,52),(10,49,35,53),(11,43,29,54),(12,44,30,55),(13,45,31,56),(14,46,32,50),(57,95,107,80),(58,96,108,81),(59,97,109,82),(60,98,110,83),(61,85,111,84),(62,86,112,71),(63,87,99,72),(64,88,100,73),(65,89,101,74),(66,90,102,75),(67,91,103,76),(68,92,104,77),(69,93,105,78),(70,94,106,79)], [(1,74,12,96),(2,82,13,90),(3,76,14,98),(4,84,8,92),(5,78,9,86),(6,72,10,94),(7,80,11,88),(15,85,33,77),(16,93,34,71),(17,87,35,79),(18,95,29,73),(19,89,30,81),(20,97,31,75),(21,91,32,83),(22,69,52,112),(23,63,53,106),(24,57,54,100),(25,65,55,108),(26,59,56,102),(27,67,50,110),(28,61,51,104),(36,103,46,60),(37,111,47,68),(38,105,48,62),(39,99,49,70),(40,107,43,64),(41,101,44,58),(42,109,45,66)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,14),(6,13),(7,12),(15,33),(16,32),(17,31),(18,30),(19,29),(20,35),(21,34),(22,50),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(42,49),(57,58),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(71,76),(72,75),(73,74),(77,84),(78,83),(79,82),(80,81),(85,92),(86,91),(87,90),(88,89),(93,98),(94,97),(95,96),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 12 | 16 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 18 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 4 | 0 | 0 |
| 0 | 0 | 25 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 18 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,16,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,16,0,0,0,0,18,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,25,0,0,0,0,4,4,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[28,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
70 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 | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D4×D7 | Q8⋊2D7 | D7×C4○D4 |
| kernel | C4⋊C4⋊26D14 | Dic7⋊4D4 | C22⋊D28 | C4⋊C4⋊7D7 | D28⋊C4 | D14.5D4 | C28⋊1D4 | C28⋊7D4 | D14⋊3Q8 | C7×C22⋊Q8 | D7×C22×C4 | C2×Q8⋊2D7 | C4×D7 | C22⋊Q8 | D14 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 4 | 6 | 9 | 3 | 3 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{26}D_{14} % in TeX
G:=Group("C4:C4:26D14"); // GroupNames label
G:=SmallGroup(448,1080);
// by ID
G=gap.SmallGroup(448,1080);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,1123,794,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations