metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.19D4, Q8.6D14, C28.15C23, D28.10C22, Dic14.9C22, Q8⋊D7⋊5C2, (C2×Q8)⋊2D7, (Q8×C14)⋊2C2, C7⋊Q16⋊5C2, C7⋊C8.3C22, C4○D28.5C2, (C2×C4).20D14, C14.54(C2×D4), (C2×C14).42D4, C7⋊4(C8.C22), C4.Dic7⋊7C2, C4.17(C7⋊D4), C4.15(C22×D7), (C7×Q8).6C22, (C2×C28).37C22, C22.11(C7⋊D4), C2.18(C2×C7⋊D4), SmallGroup(224,137)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.C23
G = < a,b,c,d | a28=b2=1, c2=d2=a14, bab=a-1, ac=ca, dad-1=a15, bc=cb, dbd-1=a21b, dcd-1=a14c >
Subgroups: 222 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, D7, C14, C14, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C8.C22, C7⋊C8, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×Q8, C7×Q8, C4.Dic7, Q8⋊D7, C7⋊Q16, C4○D28, Q8×C14, C28.C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8.C22, C7⋊D4, C22×D7, C2×C7⋊D4, C28.C23
►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)
(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 36)(30 35)(31 34)(32 33)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(74 84)(75 83)(76 82)(77 81)(78 80)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(107 112)(108 111)(109 110)
(1 65 15 79)(2 66 16 80)(3 67 17 81)(4 68 18 82)(5 69 19 83)(6 70 20 84)(7 71 21 57)(8 72 22 58)(9 73 23 59)(10 74 24 60)(11 75 25 61)(12 76 26 62)(13 77 27 63)(14 78 28 64)(29 106 43 92)(30 107 44 93)(31 108 45 94)(32 109 46 95)(33 110 47 96)(34 111 48 97)(35 112 49 98)(36 85 50 99)(37 86 51 100)(38 87 52 101)(39 88 53 102)(40 89 54 103)(41 90 55 104)(42 91 56 105)
(1 36 15 50)(2 51 16 37)(3 38 17 52)(4 53 18 39)(5 40 19 54)(6 55 20 41)(7 42 21 56)(8 29 22 43)(9 44 23 30)(10 31 24 45)(11 46 25 32)(12 33 26 47)(13 48 27 34)(14 35 28 49)(57 91 71 105)(58 106 72 92)(59 93 73 107)(60 108 74 94)(61 95 75 109)(62 110 76 96)(63 97 77 111)(64 112 78 98)(65 99 79 85)(66 86 80 100)(67 101 81 87)(68 88 82 102)(69 103 83 89)(70 90 84 104)
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), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,36)(30,35)(31,34)(32,33)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110), (1,65,15,79)(2,66,16,80)(3,67,17,81)(4,68,18,82)(5,69,19,83)(6,70,20,84)(7,71,21,57)(8,72,22,58)(9,73,23,59)(10,74,24,60)(11,75,25,61)(12,76,26,62)(13,77,27,63)(14,78,28,64)(29,106,43,92)(30,107,44,93)(31,108,45,94)(32,109,46,95)(33,110,47,96)(34,111,48,97)(35,112,49,98)(36,85,50,99)(37,86,51,100)(38,87,52,101)(39,88,53,102)(40,89,54,103)(41,90,55,104)(42,91,56,105), (1,36,15,50)(2,51,16,37)(3,38,17,52)(4,53,18,39)(5,40,19,54)(6,55,20,41)(7,42,21,56)(8,29,22,43)(9,44,23,30)(10,31,24,45)(11,46,25,32)(12,33,26,47)(13,48,27,34)(14,35,28,49)(57,91,71,105)(58,106,72,92)(59,93,73,107)(60,108,74,94)(61,95,75,109)(62,110,76,96)(63,97,77,111)(64,112,78,98)(65,99,79,85)(66,86,80,100)(67,101,81,87)(68,88,82,102)(69,103,83,89)(70,90,84,104)>;
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), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,36)(30,35)(31,34)(32,33)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110), (1,65,15,79)(2,66,16,80)(3,67,17,81)(4,68,18,82)(5,69,19,83)(6,70,20,84)(7,71,21,57)(8,72,22,58)(9,73,23,59)(10,74,24,60)(11,75,25,61)(12,76,26,62)(13,77,27,63)(14,78,28,64)(29,106,43,92)(30,107,44,93)(31,108,45,94)(32,109,46,95)(33,110,47,96)(34,111,48,97)(35,112,49,98)(36,85,50,99)(37,86,51,100)(38,87,52,101)(39,88,53,102)(40,89,54,103)(41,90,55,104)(42,91,56,105), (1,36,15,50)(2,51,16,37)(3,38,17,52)(4,53,18,39)(5,40,19,54)(6,55,20,41)(7,42,21,56)(8,29,22,43)(9,44,23,30)(10,31,24,45)(11,46,25,32)(12,33,26,47)(13,48,27,34)(14,35,28,49)(57,91,71,105)(58,106,72,92)(59,93,73,107)(60,108,74,94)(61,95,75,109)(62,110,76,96)(63,97,77,111)(64,112,78,98)(65,99,79,85)(66,86,80,100)(67,101,81,87)(68,88,82,102)(69,103,83,89)(70,90,84,104) );
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)], [(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,36),(30,35),(31,34),(32,33),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(57,73),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67),(64,66),(74,84),(75,83),(76,82),(77,81),(78,80),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(107,112),(108,111),(109,110)], [(1,65,15,79),(2,66,16,80),(3,67,17,81),(4,68,18,82),(5,69,19,83),(6,70,20,84),(7,71,21,57),(8,72,22,58),(9,73,23,59),(10,74,24,60),(11,75,25,61),(12,76,26,62),(13,77,27,63),(14,78,28,64),(29,106,43,92),(30,107,44,93),(31,108,45,94),(32,109,46,95),(33,110,47,96),(34,111,48,97),(35,112,49,98),(36,85,50,99),(37,86,51,100),(38,87,52,101),(39,88,53,102),(40,89,54,103),(41,90,55,104),(42,91,56,105)], [(1,36,15,50),(2,51,16,37),(3,38,17,52),(4,53,18,39),(5,40,19,54),(6,55,20,41),(7,42,21,56),(8,29,22,43),(9,44,23,30),(10,31,24,45),(11,46,25,32),(12,33,26,47),(13,48,27,34),(14,35,28,49),(57,91,71,105),(58,106,72,92),(59,93,73,107),(60,108,74,94),(61,95,75,109),(62,110,76,96),(63,97,77,111),(64,112,78,98),(65,99,79,85),(66,86,80,100),(67,101,81,87),(68,88,82,102),(69,103,83,89),(70,90,84,104)]])
C28.C23 is a maximal subgroup of
D28.6D4 D28.7D4 C42⋊5D14 D28.15D4 C56.44D4 C56.29D4 D28.39D4 D28.40D4 D28.29D4 D28.30D4 D7×C8.C22 D56⋊C22 C28.C24 D28.34C23 D28.35C23
C28.C23 is a maximal quotient of
C4.Dic7⋊C4 C4○D28⋊C4 (C2×C14).40D8 C4⋊C4.231D14 Q8.3Dic14 C42.56D14 Q8.1D28 C42.59D14 C14.(C4○D8) D28.37D4 C7⋊C8⋊6D4 C7⋊C8.6D4 C42.76D14 C42.77D14 C28⋊5SD16 C42.80D14 D28⋊6Q8 C42.82D14 C28⋊Q16 Dic14⋊6Q8 (Q8×C14)⋊6C4 (C7×Q8)⋊13D4 (C2×C14)⋊8Q16
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 28 | 2 | 2 | 4 | 4 | 28 | 2 | 2 | 2 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8.C22 | C28.C23 |
| kernel | C28.C23 | C4.Dic7 | Q8⋊D7 | C7⋊Q16 | C4○D28 | Q8×C14 | C28 | C2×C14 | C2×Q8 | C2×C4 | Q8 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 1 | 6 |
Matrix representation of C28.C23 ►in GL4(𝔽113) generated by
| 16 | 2 | 111 | 24 |
| 50 | 27 | 24 | 111 |
| 22 | 76 | 86 | 111 |
| 101 | 22 | 63 | 97 |
| 103 | 89 | 0 | 0 |
| 103 | 10 | 0 | 0 |
| 94 | 100 | 112 | 0 |
| 40 | 100 | 89 | 1 |
| 19 | 55 | 90 | 81 |
| 91 | 14 | 19 | 90 |
| 16 | 37 | 99 | 58 |
| 75 | 16 | 22 | 94 |
| 60 | 65 | 16 | 22 |
| 84 | 89 | 46 | 16 |
| 38 | 84 | 24 | 48 |
| 69 | 38 | 29 | 53 |
G:=sub<GL(4,GF(113))| [16,50,22,101,2,27,76,22,111,24,86,63,24,111,111,97],[103,103,94,40,89,10,100,100,0,0,112,89,0,0,0,1],[19,91,16,75,55,14,37,16,90,19,99,22,81,90,58,94],[60,84,38,69,65,89,84,38,16,46,24,29,22,16,48,53] >;
C28.C23 in GAP, Magma, Sage, TeX
C_{28}.C_2^3 % in TeX
G:=Group("C28.C2^3"); // GroupNames label
G:=SmallGroup(224,137);
// by ID
G=gap.SmallGroup(224,137);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,86,579,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^2=d^2=a^14,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^15,b*c=c*b,d*b*d^-1=a^21*b,d*c*d^-1=a^14*c>;
// generators/relations