direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D7×M4(2), C56⋊7C23, C28.68C24, (C2×C8)⋊29D14, C7⋊C8⋊12C23, C8⋊7(C22×D7), (C2×C56)⋊23C22, (C8×D7)⋊21C22, C14⋊2(C2×M4(2)), (C23×D7).9C4, C4.67(C23×D7), C23.58(C4×D7), C7⋊2(C22×M4(2)), C8⋊D7⋊17C22, (C14×M4(2))⋊8C2, C28.91(C22×C4), C14.31(C23×C4), (C4×D7).40C23, (C2×C28).881C23, D14.26(C22×C4), (C22×C4).373D14, C4.Dic7⋊25C22, (C7×M4(2))⋊24C22, Dic7.27(C22×C4), (C22×Dic7).18C4, (C22×C28).263C22, (D7×C2×C8)⋊28C2, (C2×C4×D7).10C4, C4.122(C2×C4×D7), (C2×C7⋊C8)⋊47C22, (C2×C8⋊D7)⋊26C2, (D7×C22×C4).8C2, C22.76(C2×C4×D7), C2.32(D7×C22×C4), (C4×D7).30(C2×C4), (C2×C4).162(C4×D7), (C2×C28).130(C2×C4), (C2×C4.Dic7)⋊24C2, (C2×C4×D7).253C22, (C22×C14).77(C2×C4), (C2×C14).24(C22×C4), (C22×D7).68(C2×C4), (C2×C4).604(C22×D7), (C2×Dic7).105(C2×C4), SmallGroup(448,1196)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1124 in 298 conjugacy classes, 159 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×20], C7, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], C23, C23 [×10], D7 [×4], D7 [×2], C14, C14 [×2], C14 [×2], C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×13], C24, Dic7 [×4], C28 [×2], C28 [×2], D14 [×8], D14 [×10], C2×C14, C2×C14 [×2], C2×C14 [×2], C22×C8 [×2], C2×M4(2), C2×M4(2) [×11], C23×C4, C7⋊C8 [×4], C56 [×4], C4×D7 [×16], C2×Dic7 [×2], C2×Dic7 [×4], C2×C28 [×2], C2×C28 [×4], C22×D7 [×2], C22×D7 [×4], C22×D7 [×4], C22×C14, C22×M4(2), C8×D7 [×8], C8⋊D7 [×8], C2×C7⋊C8 [×2], C4.Dic7 [×4], C2×C56 [×2], C7×M4(2) [×4], C2×C4×D7 [×4], C2×C4×D7 [×8], C22×Dic7, C22×C28, C23×D7, D7×C2×C8 [×2], C2×C8⋊D7 [×2], D7×M4(2) [×8], C2×C4.Dic7, C14×M4(2), D7×C22×C4, C2×D7×M4(2)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, M4(2) [×4], C22×C4 [×14], C24, D14 [×7], C2×M4(2) [×6], C23×C4, C4×D7 [×4], C22×D7 [×7], C22×M4(2), C2×C4×D7 [×6], C23×D7, D7×M4(2) [×2], D7×C22×C4, C2×D7×M4(2)
Generators and relations
G = < a,b,c,d,e | a2=b7=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >
►On 112 points
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(33 109)(34 110)(35 111)(36 112)(37 105)(38 106)(39 107)(40 108)(41 96)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 81)(73 100)(74 101)(75 102)(76 103)(77 104)(78 97)(79 98)(80 99)
(1 102 21 12 86 47 36)(2 103 22 13 87 48 37)(3 104 23 14 88 41 38)(4 97 24 15 81 42 39)(5 98 17 16 82 43 40)(6 99 18 9 83 44 33)(7 100 19 10 84 45 34)(8 101 20 11 85 46 35)(25 76 66 57 54 95 105)(26 77 67 58 55 96 106)(27 78 68 59 56 89 107)(28 79 69 60 49 90 108)(29 80 70 61 50 91 109)(30 73 71 62 51 92 110)(31 74 72 63 52 93 111)(32 75 65 64 53 94 112)
(1 108)(2 109)(3 110)(4 111)(5 112)(6 105)(7 106)(8 107)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 81)(89 101)(90 102)(91 103)(92 104)(93 97)(94 98)(95 99)(96 100)
(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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)
G:=sub<Sym(112)| (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(33,109)(34,110)(35,111)(36,112)(37,105)(38,106)(39,107)(40,108)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,81)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99), (1,102,21,12,86,47,36)(2,103,22,13,87,48,37)(3,104,23,14,88,41,38)(4,97,24,15,81,42,39)(5,98,17,16,82,43,40)(6,99,18,9,83,44,33)(7,100,19,10,84,45,34)(8,101,20,11,85,46,35)(25,76,66,57,54,95,105)(26,77,67,58,55,96,106)(27,78,68,59,56,89,107)(28,79,69,60,49,90,108)(29,80,70,61,50,91,109)(30,73,71,62,51,92,110)(31,74,72,63,52,93,111)(32,75,65,64,53,94,112), (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,81)(89,101)(90,102)(91,103)(92,104)(93,97)(94,98)(95,99)(96,100), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111)>;
G:=Group( (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(33,109)(34,110)(35,111)(36,112)(37,105)(38,106)(39,107)(40,108)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,81)(73,100)(74,101)(75,102)(76,103)(77,104)(78,97)(79,98)(80,99), (1,102,21,12,86,47,36)(2,103,22,13,87,48,37)(3,104,23,14,88,41,38)(4,97,24,15,81,42,39)(5,98,17,16,82,43,40)(6,99,18,9,83,44,33)(7,100,19,10,84,45,34)(8,101,20,11,85,46,35)(25,76,66,57,54,95,105)(26,77,67,58,55,96,106)(27,78,68,59,56,89,107)(28,79,69,60,49,90,108)(29,80,70,61,50,91,109)(30,73,71,62,51,92,110)(31,74,72,63,52,93,111)(32,75,65,64,53,94,112), (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,81)(89,101)(90,102)(91,103)(92,104)(93,97)(94,98)(95,99)(96,100), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111) );
G=PermutationGroup([(1,32),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,69),(18,70),(19,71),(20,72),(21,65),(22,66),(23,67),(24,68),(33,109),(34,110),(35,111),(36,112),(37,105),(38,106),(39,107),(40,108),(41,96),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,81),(73,100),(74,101),(75,102),(76,103),(77,104),(78,97),(79,98),(80,99)], [(1,102,21,12,86,47,36),(2,103,22,13,87,48,37),(3,104,23,14,88,41,38),(4,97,24,15,81,42,39),(5,98,17,16,82,43,40),(6,99,18,9,83,44,33),(7,100,19,10,84,45,34),(8,101,20,11,85,46,35),(25,76,66,57,54,95,105),(26,77,67,58,55,96,106),(27,78,68,59,56,89,107),(28,79,69,60,49,90,108),(29,80,70,61,50,91,109),(30,73,71,62,51,92,110),(31,74,72,63,52,93,111),(32,75,65,64,53,94,112)], [(1,108),(2,109),(3,110),(4,111),(5,112),(6,105),(7,106),(8,107),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,53),(18,54),(19,55),(20,56),(21,49),(22,50),(23,51),(24,52),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,81),(89,101),(90,102),(91,103),(92,104),(93,97),(94,98),(95,99),(96,100)], [(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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(57,61),(59,63),(66,70),(68,72),(74,78),(76,80),(81,85),(83,87),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111)])
Matrix representation ►G ⊆ GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 88 | 2 |
| 0 | 0 | 0 | 112 | 104 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 88 | 59 |
| 0 | 0 | 0 | 112 | 25 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 98 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,112,0,0,0,2,104],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,112,0,0,0,59,25],[1,0,0,0,0,0,0,98,0,0,0,112,0,0,0,0,0,0,98,0,0,0,0,0,98],[112,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1] >;
100 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 | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| 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 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | M4(2) | D14 | D14 | D14 | C4×D7 | C4×D7 | D7×M4(2) |
| kernel | C2×D7×M4(2) | D7×C2×C8 | C2×C8⋊D7 | D7×M4(2) | C2×C4.Dic7 | C14×M4(2) | D7×C22×C4 | C2×C4×D7 | C22×Dic7 | C23×D7 | C2×M4(2) | D14 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 12 | 2 | 2 | 3 | 8 | 6 | 12 | 3 | 18 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_7\times M_{4(2)} % in TeX
G:=Group("C2xD7xM4(2)"); // GroupNames label
G:=SmallGroup(448,1196);
// by ID
G=gap.SmallGroup(448,1196);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,297,80,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations