direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.1D14, C24.11D14, C23.15D28, (C23×D7)⋊4C4, C14⋊1(C23⋊C4), C22⋊C4⋊36D14, C23.17(C4×D7), (C22×Dic7)⋊5C4, C22.14(C2×D28), (C22×C14).66D4, C23.74(C7⋊D4), C23.D7⋊42C22, C23.72(C22×D7), C22.44(D14⋊C4), (C23×C14).37C22, (C22×C14).111C23, C7⋊2(C2×C23⋊C4), (C2×C7⋊D4)⋊3C4, C2.8(C2×D14⋊C4), (C2×C22⋊C4)⋊1D7, (C14×C22⋊C4)⋊1C2, C22.18(C2×C4×D7), (C2×Dic7)⋊2(C2×C4), (C22×D7)⋊2(C2×C4), (C2×C23.D7)⋊1C2, (C2×C14).433(C2×D4), C14.35(C2×C22⋊C4), (C22×C7⋊D4).1C2, C22.26(C2×C7⋊D4), (C7×C22⋊C4)⋊44C22, (C2×C14).12(C22×C4), (C22×C14).52(C2×C4), (C2×C7⋊D4).83C22, (C2×C14).56(C22⋊C4), SmallGroup(448,488)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.1D14
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 1172 in 210 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C2×C23⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C22×Dic7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C23.1D14, C2×C23.D7, C14×C22⋊C4, C22×C7⋊D4, C2×C23.1D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C23⋊C4, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C2×C23⋊C4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C23.1D14, C2×D14⋊C4, C2×C23.1D14
►On 112 points
Generators in S112
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)
(1 80)(3 82)(5 84)(7 58)(9 60)(11 62)(13 64)(15 66)(17 68)(19 70)(21 72)(23 74)(25 76)(27 78)(30 104)(32 106)(34 108)(36 110)(38 112)(40 86)(42 88)(44 90)(46 92)(48 94)(50 96)(52 98)(54 100)(56 102)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)
(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 96 80 50)(2 95)(3 48 82 94)(4 47)(5 92 84 46)(6 91)(7 44 58 90)(8 43)(9 88 60 42)(10 87)(11 40 62 86)(12 39)(13 112 64 38)(14 111)(15 36 66 110)(16 35)(17 108 68 34)(18 107)(19 32 70 106)(20 31)(21 104 72 30)(22 103)(23 56 74 102)(24 55)(25 100 76 54)(26 99)(27 52 78 98)(28 51)(29 73)(33 69)(37 65)(41 61)(45 57)(49 81)(53 77)(59 89)(63 85)(67 109)(71 105)(75 101)(79 97)(83 93)
G:=sub<Sym(112)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94), (1,80)(3,82)(5,84)(7,58)(9,60)(11,62)(13,64)(15,66)(17,68)(19,70)(21,72)(23,74)(25,76)(27,78)(30,104)(32,106)(34,108)(36,110)(38,112)(40,86)(42,88)(44,90)(46,92)(48,94)(50,96)(52,98)(54,100)(56,102), (1,80)(2,81)(3,82)(4,83)(5,84)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102), (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), (1,96,80,50)(2,95)(3,48,82,94)(4,47)(5,92,84,46)(6,91)(7,44,58,90)(8,43)(9,88,60,42)(10,87)(11,40,62,86)(12,39)(13,112,64,38)(14,111)(15,36,66,110)(16,35)(17,108,68,34)(18,107)(19,32,70,106)(20,31)(21,104,72,30)(22,103)(23,56,74,102)(24,55)(25,100,76,54)(26,99)(27,52,78,98)(28,51)(29,73)(33,69)(37,65)(41,61)(45,57)(49,81)(53,77)(59,89)(63,85)(67,109)(71,105)(75,101)(79,97)(83,93)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94), (1,80)(3,82)(5,84)(7,58)(9,60)(11,62)(13,64)(15,66)(17,68)(19,70)(21,72)(23,74)(25,76)(27,78)(30,104)(32,106)(34,108)(36,110)(38,112)(40,86)(42,88)(44,90)(46,92)(48,94)(50,96)(52,98)(54,100)(56,102), (1,80)(2,81)(3,82)(4,83)(5,84)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102), (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), (1,96,80,50)(2,95)(3,48,82,94)(4,47)(5,92,84,46)(6,91)(7,44,58,90)(8,43)(9,88,60,42)(10,87)(11,40,62,86)(12,39)(13,112,64,38)(14,111)(15,36,66,110)(16,35)(17,108,68,34)(18,107)(19,32,70,106)(20,31)(21,104,72,30)(22,103)(23,56,74,102)(24,55)(25,100,76,54)(26,99)(27,52,78,98)(28,51)(29,73)(33,69)(37,65)(41,61)(45,57)(49,81)(53,77)(59,89)(63,85)(67,109)(71,105)(75,101)(79,97)(83,93) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(57,95),(58,96),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,111),(74,112),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(81,91),(82,92),(83,93),(84,94)], [(1,80),(3,82),(5,84),(7,58),(9,60),(11,62),(13,64),(15,66),(17,68),(19,70),(21,72),(23,74),(25,76),(27,78),(30,104),(32,106),(34,108),(36,110),(38,112),(40,86),(42,88),(44,90),(46,92),(48,94),(50,96),(52,98),(54,100),(56,102)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,99),(54,100),(55,101),(56,102)], [(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)], [(1,96,80,50),(2,95),(3,48,82,94),(4,47),(5,92,84,46),(6,91),(7,44,58,90),(8,43),(9,88,60,42),(10,87),(11,40,62,86),(12,39),(13,112,64,38),(14,111),(15,36,66,110),(16,35),(17,108,68,34),(18,107),(19,32,70,106),(20,31),(21,104,72,30),(22,103),(23,56,74,102),(24,55),(25,100,76,54),(26,99),(27,52,78,98),(28,51),(29,73),(33,69),(37,65),(41,61),(45,57),(49,81),(53,77),(59,89),(63,85),(67,109),(71,105),(75,101),(79,97),(83,93)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | C23⋊C4 | C23.1D14 |
| kernel | C2×C23.1D14 | C23.1D14 | C2×C23.D7 | C14×C22⋊C4 | C22×C7⋊D4 | C22×Dic7 | C2×C7⋊D4 | C23×D7 | C22×C14 | C2×C22⋊C4 | C22⋊C4 | C24 | C23 | C23 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 3 | 6 | 3 | 12 | 12 | 12 | 2 | 12 |
Matrix representation of C2×C23.1D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 27 | 17 | 1 | 0 |
| 0 | 0 | 12 | 20 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 27 | 6 |
| 0 | 0 | 15 | 22 | 23 | 16 |
| 0 | 0 | 24 | 2 | 20 | 15 |
| 0 | 0 | 27 | 1 | 14 | 7 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 27 | 0 | 0 |
| 0 | 0 | 13 | 24 | 0 | 0 |
| 0 | 0 | 6 | 19 | 1 | 26 |
| 0 | 0 | 8 | 23 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,27,12,0,0,0,28,17,20,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,9,15,24,27,0,0,14,22,2,1,0,0,27,23,20,14,0,0,6,16,15,7],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,5,13,6,8,0,0,27,24,19,23,0,0,0,0,1,0,0,0,0,0,26,28] >;
C2×C23.1D14 in GAP, Magma, Sage, TeX
C_2\times C_2^3._1D_{14} % in TeX
G:=Group("C2xC2^3.1D14"); // GroupNames label
G:=SmallGroup(448,488);
// by ID
G=gap.SmallGroup(448,488);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,1123,438,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations