metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.76C24, C7⋊C8.34C23, C4○D4.45D14, (D4×C14).13C4, C7⋊3(Q8○M4(2)), D4.Dic7⋊9C2, (Q8×C14).13C4, C4○D4.2Dic7, (C2×Q8).8Dic7, D4.9(C2×Dic7), C4.75(C23×D7), C14.49(C23×C4), C28.98(C22×C4), Q8.10(C2×Dic7), (C2×D4).10Dic7, (C2×C28).554C23, (C22×C4).280D14, C4.Dic7⋊35C22, C2.11(C23×Dic7), C23.11(C2×Dic7), C4.20(C22×Dic7), C22.2(C22×Dic7), (C22×C28).289C22, (C2×C7⋊C8)⋊21C22, (C2×C4○D4).9D7, (C7×C4○D4).4C4, (C7×D4).26(C2×C4), (C7×Q8).28(C2×C4), (C2×C28).136(C2×C4), (C14×C4○D4).10C2, (C2×C4.Dic7)⋊29C2, (C2×C4).31(C2×Dic7), (C22×C14).81(C2×C4), (C2×C14).29(C22×C4), (C7×C4○D4).49C22, (C2×C4).635(C22×D7), SmallGroup(448,1272)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 596 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C7, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C14, C14 [×7], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C28 [×2], C28 [×6], C2×C14, C2×C14 [×6], C2×C14 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C7⋊C8 [×8], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×C14 [×3], Q8○M4(2), C2×C7⋊C8 [×12], C4.Dic7 [×16], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C2×C4.Dic7 [×6], D4.Dic7 [×8], C14×C4○D4, C28.76C24
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, Dic7 [×8], D14 [×7], C23×C4, C2×Dic7 [×28], C22×D7 [×7], Q8○M4(2), C22×Dic7 [×14], C23×D7, C23×Dic7, C28.76C24
Generators and relations
G = < a,b,c,d,e | a28=c2=d2=e2=1, b2=a21, bab-1=a13, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a14b, dcd=a14c, ce=ec, de=ed >
►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 66 22 59 15 80 8 73)(2 79 23 72 16 65 9 58)(3 64 24 57 17 78 10 71)(4 77 25 70 18 63 11 84)(5 62 26 83 19 76 12 69)(6 75 27 68 20 61 13 82)(7 60 28 81 21 74 14 67)(29 107 50 100 43 93 36 86)(30 92 51 85 44 106 37 99)(31 105 52 98 45 91 38 112)(32 90 53 111 46 104 39 97)(33 103 54 96 47 89 40 110)(34 88 55 109 48 102 41 95)(35 101 56 94 49 87 42 108)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 85)(83 86)(84 87)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
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), (1,66,22,59,15,80,8,73)(2,79,23,72,16,65,9,58)(3,64,24,57,17,78,10,71)(4,77,25,70,18,63,11,84)(5,62,26,83,19,76,12,69)(6,75,27,68,20,61,13,82)(7,60,28,81,21,74,14,67)(29,107,50,100,43,93,36,86)(30,92,51,85,44,106,37,99)(31,105,52,98,45,91,38,112)(32,90,53,111,46,104,39,97)(33,103,54,96,47,89,40,110)(34,88,55,109,48,102,41,95)(35,101,56,94,49,87,42,108), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,85)(83,86)(84,87), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)>;
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), (1,66,22,59,15,80,8,73)(2,79,23,72,16,65,9,58)(3,64,24,57,17,78,10,71)(4,77,25,70,18,63,11,84)(5,62,26,83,19,76,12,69)(6,75,27,68,20,61,13,82)(7,60,28,81,21,74,14,67)(29,107,50,100,43,93,36,86)(30,92,51,85,44,106,37,99)(31,105,52,98,45,91,38,112)(32,90,53,111,46,104,39,97)(33,103,54,96,47,89,40,110)(34,88,55,109,48,102,41,95)(35,101,56,94,49,87,42,108), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,85)(83,86)(84,87), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112) );
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)], [(1,66,22,59,15,80,8,73),(2,79,23,72,16,65,9,58),(3,64,24,57,17,78,10,71),(4,77,25,70,18,63,11,84),(5,62,26,83,19,76,12,69),(6,75,27,68,20,61,13,82),(7,60,28,81,21,74,14,67),(29,107,50,100,43,93,36,86),(30,92,51,85,44,106,37,99),(31,105,52,98,45,91,38,112),(32,90,53,111,46,104,39,97),(33,103,54,96,47,89,40,110),(34,88,55,109,48,102,41,95),(35,101,56,94,49,87,42,108)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,85),(83,86),(84,87)], [(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 99 | 0 | 0 | 0 |
| 0 | 99 | 0 | 0 |
| 0 | 0 | 105 | 0 |
| 0 | 0 | 0 | 105 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [99,0,0,0,0,99,0,0,0,0,105,0,0,0,0,105],[0,0,98,0,0,0,0,98,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112] >;
94 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 4A | 4B | 4C | ··· | 4I | 7A | 7B | 7C | 8A | ··· | 8P | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | D14 | Dic7 | Dic7 | Dic7 | D14 | Q8○M4(2) | C28.76C24 |
| kernel | C28.76C24 | C2×C4.Dic7 | D4.Dic7 | C14×C4○D4 | D4×C14 | Q8×C14 | C7×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C7 | C1 |
| # reps | 1 | 6 | 8 | 1 | 6 | 2 | 8 | 3 | 9 | 9 | 3 | 12 | 12 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{28}._{76}C_2^4 % in TeX
G:=Group("C28.76C2^4"); // GroupNames label
G:=SmallGroup(448,1272);
// by ID
G=gap.SmallGroup(448,1272);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,387,1123,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^28=c^2=d^2=e^2=1,b^2=a^21,b*a*b^-1=a^13,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^14*b,d*c*d=a^14*c,c*e=e*c,d*e=e*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀