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), C4⋊D28⋊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×C14).171C24, (C2×C28).598C23, 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, (C23×D7).109C22, (C22×D7).193C23, (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
Generators and relations for C4⋊C4⋊26D14
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 >
Subgroups: 1692 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×D7, C22×C14, C22.19C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q8⋊2D7, C22×Dic7, C2×C7⋊D4, C22×C28, Q8×C14, C23×D7, Dic7⋊4D4, C22⋊D28, C4⋊C4⋊7D7, D28⋊C4, D14.5D4, C4⋊D28, C28⋊7D4, D14⋊3Q8, C7×C22⋊Q8, D7×C22×C4, C2×Q8⋊2D7, C4⋊C4⋊26D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, C22×D7, C22.19C24, D4×D7, Q8⋊2D7, C23×D7, C2×D4×D7, C2×Q8⋊2D7, D7×C4○D4, C4⋊C4⋊26D14
►On 112 points
(1 44 15 9)(2 45 16 10)(3 46 17 11)(4 47 18 12)(5 48 19 13)(6 49 20 14)(7 43 21 8)(22 55 32 40)(23 56 33 41)(24 50 34 42)(25 51 35 36)(26 52 29 37)(27 53 30 38)(28 54 31 39)(57 102 94 83)(58 103 95 84)(59 104 96 71)(60 105 97 72)(61 106 98 73)(62 107 85 74)(63 108 86 75)(64 109 87 76)(65 110 88 77)(66 111 89 78)(67 112 90 79)(68 99 91 80)(69 100 92 81)(70 101 93 82)
(1 70 32 86)(2 64 33 94)(3 58 34 88)(4 66 35 96)(5 60 29 90)(6 68 30 98)(7 62 31 92)(8 107 54 81)(9 101 55 75)(10 109 56 83)(11 103 50 77)(12 111 51 71)(13 105 52 79)(14 99 53 73)(15 93 22 63)(16 87 23 57)(17 95 24 65)(18 89 25 59)(19 97 26 67)(20 91 27 61)(21 85 28 69)(36 104 47 78)(37 112 48 72)(38 106 49 80)(39 100 43 74)(40 108 44 82)(41 102 45 76)(42 110 46 84)
(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 31)(2 30)(3 29)(4 35)(5 34)(6 33)(7 32)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 56)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(36 47)(37 46)(38 45)(39 44)(40 43)(41 49)(42 48)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(69 70)(71 78)(72 77)(73 76)(74 75)(79 84)(80 83)(81 82)(85 86)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(112)| (1,44,15,9)(2,45,16,10)(3,46,17,11)(4,47,18,12)(5,48,19,13)(6,49,20,14)(7,43,21,8)(22,55,32,40)(23,56,33,41)(24,50,34,42)(25,51,35,36)(26,52,29,37)(27,53,30,38)(28,54,31,39)(57,102,94,83)(58,103,95,84)(59,104,96,71)(60,105,97,72)(61,106,98,73)(62,107,85,74)(63,108,86,75)(64,109,87,76)(65,110,88,77)(66,111,89,78)(67,112,90,79)(68,99,91,80)(69,100,92,81)(70,101,93,82), (1,70,32,86)(2,64,33,94)(3,58,34,88)(4,66,35,96)(5,60,29,90)(6,68,30,98)(7,62,31,92)(8,107,54,81)(9,101,55,75)(10,109,56,83)(11,103,50,77)(12,111,51,71)(13,105,52,79)(14,99,53,73)(15,93,22,63)(16,87,23,57)(17,95,24,65)(18,89,25,59)(19,97,26,67)(20,91,27,61)(21,85,28,69)(36,104,47,78)(37,112,48,72)(38,106,49,80)(39,100,43,74)(40,108,44,82)(41,102,45,76)(42,110,46,84), (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), (1,31)(2,30)(3,29)(4,35)(5,34)(6,33)(7,32)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,56)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(36,47)(37,46)(38,45)(39,44)(40,43)(41,49)(42,48)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,86)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,44,15,9)(2,45,16,10)(3,46,17,11)(4,47,18,12)(5,48,19,13)(6,49,20,14)(7,43,21,8)(22,55,32,40)(23,56,33,41)(24,50,34,42)(25,51,35,36)(26,52,29,37)(27,53,30,38)(28,54,31,39)(57,102,94,83)(58,103,95,84)(59,104,96,71)(60,105,97,72)(61,106,98,73)(62,107,85,74)(63,108,86,75)(64,109,87,76)(65,110,88,77)(66,111,89,78)(67,112,90,79)(68,99,91,80)(69,100,92,81)(70,101,93,82), (1,70,32,86)(2,64,33,94)(3,58,34,88)(4,66,35,96)(5,60,29,90)(6,68,30,98)(7,62,31,92)(8,107,54,81)(9,101,55,75)(10,109,56,83)(11,103,50,77)(12,111,51,71)(13,105,52,79)(14,99,53,73)(15,93,22,63)(16,87,23,57)(17,95,24,65)(18,89,25,59)(19,97,26,67)(20,91,27,61)(21,85,28,69)(36,104,47,78)(37,112,48,72)(38,106,49,80)(39,100,43,74)(40,108,44,82)(41,102,45,76)(42,110,46,84), (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), (1,31)(2,30)(3,29)(4,35)(5,34)(6,33)(7,32)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,56)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(36,47)(37,46)(38,45)(39,44)(40,43)(41,49)(42,48)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,86)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([[(1,44,15,9),(2,45,16,10),(3,46,17,11),(4,47,18,12),(5,48,19,13),(6,49,20,14),(7,43,21,8),(22,55,32,40),(23,56,33,41),(24,50,34,42),(25,51,35,36),(26,52,29,37),(27,53,30,38),(28,54,31,39),(57,102,94,83),(58,103,95,84),(59,104,96,71),(60,105,97,72),(61,106,98,73),(62,107,85,74),(63,108,86,75),(64,109,87,76),(65,110,88,77),(66,111,89,78),(67,112,90,79),(68,99,91,80),(69,100,92,81),(70,101,93,82)], [(1,70,32,86),(2,64,33,94),(3,58,34,88),(4,66,35,96),(5,60,29,90),(6,68,30,98),(7,62,31,92),(8,107,54,81),(9,101,55,75),(10,109,56,83),(11,103,50,77),(12,111,51,71),(13,105,52,79),(14,99,53,73),(15,93,22,63),(16,87,23,57),(17,95,24,65),(18,89,25,59),(19,97,26,67),(20,91,27,61),(21,85,28,69),(36,104,47,78),(37,112,48,72),(38,106,49,80),(39,100,43,74),(40,108,44,82),(41,102,45,76),(42,110,46,84)], [(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)], [(1,31),(2,30),(3,29),(4,35),(5,34),(6,33),(7,32),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,56),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(36,47),(37,46),(38,45),(39,44),(40,43),(41,49),(42,48),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(69,70),(71,78),(72,77),(73,76),(74,75),(79,84),(80,83),(81,82),(85,86),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108)]])
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 | C4⋊D28 | 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 |
Matrix representation of C4⋊C4⋊26D14 ►in 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] >;
C4⋊C4⋊26D14 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