metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.23D4, (C2×Q8)⋊4D7, (Q8×C14)⋊4C2, D14⋊C4⋊16C2, (C4×Dic7)⋊7C2, (C2×D28).9C2, C14.58(C2×D4), (C2×C4).57D14, C7⋊4(C4.4D4), C4.11(C7⋊D4), C14.37(C4○D4), (C2×C14).59C23, (C2×C28).40C22, C2.9(Q8⋊2D7), C22.65(C22×D7), (C2×Dic7).41C22, (C22×D7).13C22, C2.22(C2×C7⋊D4), SmallGroup(224,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.23D4
G = < a,b,c | a28=b4=c2=1, bab-1=a13, cac=a-1, cbc=a14b-1 >
Subgroups: 350 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4.4D4, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C4×Dic7, D14⋊C4, C2×D28, Q8×C14, C28.23D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C7⋊D4, C22×D7, Q8⋊2D7, C2×C7⋊D4, C28.23D4
►On 112 points
(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 73 42 91)(2 58 43 104)(3 71 44 89)(4 84 45 102)(5 69 46 87)(6 82 47 100)(7 67 48 85)(8 80 49 98)(9 65 50 111)(10 78 51 96)(11 63 52 109)(12 76 53 94)(13 61 54 107)(14 74 55 92)(15 59 56 105)(16 72 29 90)(17 57 30 103)(18 70 31 88)(19 83 32 101)(20 68 33 86)(21 81 34 99)(22 66 35 112)(23 79 36 97)(24 64 37 110)(25 77 38 95)(26 62 39 108)(27 75 40 93)(28 60 41 106)
(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 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(57 93)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)
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), (1,73,42,91)(2,58,43,104)(3,71,44,89)(4,84,45,102)(5,69,46,87)(6,82,47,100)(7,67,48,85)(8,80,49,98)(9,65,50,111)(10,78,51,96)(11,63,52,109)(12,76,53,94)(13,61,54,107)(14,74,55,92)(15,59,56,105)(16,72,29,90)(17,57,30,103)(18,70,31,88)(19,83,32,101)(20,68,33,86)(21,81,34,99)(22,66,35,112)(23,79,36,97)(24,64,37,110)(25,77,38,95)(26,62,39,108)(27,75,40,93)(28,60,41,106), (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)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)>;
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), (1,73,42,91)(2,58,43,104)(3,71,44,89)(4,84,45,102)(5,69,46,87)(6,82,47,100)(7,67,48,85)(8,80,49,98)(9,65,50,111)(10,78,51,96)(11,63,52,109)(12,76,53,94)(13,61,54,107)(14,74,55,92)(15,59,56,105)(16,72,29,90)(17,57,30,103)(18,70,31,88)(19,83,32,101)(20,68,33,86)(21,81,34,99)(22,66,35,112)(23,79,36,97)(24,64,37,110)(25,77,38,95)(26,62,39,108)(27,75,40,93)(28,60,41,106), (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)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94) );
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)], [(1,73,42,91),(2,58,43,104),(3,71,44,89),(4,84,45,102),(5,69,46,87),(6,82,47,100),(7,67,48,85),(8,80,49,98),(9,65,50,111),(10,78,51,96),(11,63,52,109),(12,76,53,94),(13,61,54,107),(14,74,55,92),(15,59,56,105),(16,72,29,90),(17,57,30,103),(18,70,31,88),(19,83,32,101),(20,68,33,86),(21,81,34,99),(22,66,35,112),(23,79,36,97),(24,64,37,110),(25,77,38,95),(26,62,39,108),(27,75,40,93),(28,60,41,106)], [(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),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(57,93),(58,92),(59,91),(60,90),(61,89),(62,88),(63,87),(64,86),(65,85),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94)]])
C28.23D4 is a maximal subgroup of
(C2×C4).D28 D28.5D4 Q8⋊C4⋊D7 Dic14.11D4 Q8⋊Dic7⋊C2 D28.12D4 (C7×D4).D4 C56.43D4 C56⋊9D4 (C2×Q16)⋊D7 C56.37D4 C56.28D4 D28.39D4 2- 1+4⋊D7 C42.122D14 C42.131D14 C42.133D14 C42.136D14 C22⋊Q8⋊25D7 D28⋊21D4 Dic14⋊22D4 C14.532+ 1+4 C14.222- 1+4 C14.242- 1+4 C14.562+ 1+4 C14.262- 1+4 C42.138D14 D7×C4.4D4 C42⋊18D14 C42⋊20D14 C42.143D14 C42⋊22D14 C42.171D14 C42.240D14 C42.177D14 C42.178D14 C42.179D14 C42.180D14 C14.442- 1+4 C14.452- 1+4 C14.1452+ 1+4 C14.1462+ 1+4 (C2×C28)⋊17D4
C28.23D4 is a maximal quotient of
(C4×Dic7)⋊9C4 (C2×C28).55D4 (C2×D28)⋊10C4 D14⋊C4⋊7C4 (C2×C4)⋊3D28 (C2×C28).290D4 C42.70D14 C42.216D14 C42.71D14 C28.D8 C42.82D14 C28.11Q16 (Q8×C14)⋊7C4 (C22×Q8)⋊D7
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | C7⋊D4 | Q8⋊2D7 |
| kernel | C28.23D4 | C4×Dic7 | D14⋊C4 | C2×D28 | Q8×C14 | C28 | C2×Q8 | C14 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 3 | 4 | 9 | 12 | 6 |
Matrix representation of C28.23D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 26 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 6 | 19 | 0 | 0 | 0 | 0 |
| 24 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 27 | 28 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,10,0,0,0,0,26,0,0,0,0,0,0,0,12,0,0,0,0,0,12,17],[6,24,0,0,0,0,19,23,0,0,0,0,0,0,0,10,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,28],[1,7,0,0,0,0,0,28,0,0,0,0,0,0,0,19,0,0,0,0,26,0,0,0,0,0,0,0,1,27,0,0,0,0,0,28] >;
C28.23D4 in GAP, Magma, Sage, TeX
C_{28}._{23}D_4 % in TeX
G:=Group("C28.23D4"); // GroupNames label
G:=SmallGroup(224,142);
// by ID
G=gap.SmallGroup(224,142);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,218,188,86,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=a^14*b^-1>;
// generators/relations