metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.38D14, C14.872+ 1+4, (D4×C14)⋊12C4, D4⋊6(C2×Dic7), (D4×Dic7)⋊37C2, (C2×D4)⋊11Dic7, C23⋊3(C2×Dic7), (C2×D4).251D14, C28.94(C22×C4), C14.45(C23×C4), C4⋊Dic7⋊76C22, (C22×D4).12D7, C2.5(D4⋊6D14), C2.7(C23×Dic7), (C2×C28).541C23, (C2×C14).293C24, C7⋊4(C22.11C24), (C4×Dic7)⋊40C22, (C22×C4).270D14, C23.D7⋊59C22, C4.17(C22×Dic7), C22.45(C23×D7), (D4×C14).270C22, (C23×C14).75C22, C23.204(C22×D7), C23.21D14⋊32C2, C22.1(C22×Dic7), (C22×C14).229C23, (C22×C28).274C22, (C2×Dic7).283C23, (C22×Dic7)⋊31C22, (D4×C2×C14).9C2, (C2×C28)⋊15(C2×C4), (C7×D4)⋊20(C2×C4), (C2×C4)⋊4(C2×Dic7), (C22×C14)⋊12(C2×C4), (C2×C23.D7)⋊26C2, (C2×C14).27(C22×C4), (C2×C4).624(C22×D7), SmallGroup(448,1251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.38D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 1108 in 338 conjugacy classes, 191 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C22.11C24, C4×Dic7, C4⋊Dic7, C23.D7, C22×Dic7, C22×C28, D4×C14, C23×C14, C23.21D14, D4×Dic7, C2×C23.D7, D4×C2×C14, C24.38D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, Dic7, D14, C23×C4, 2+ 1+4, C2×Dic7, C22×D7, C22.11C24, C22×Dic7, C23×D7, D4⋊6D14, C23×Dic7, C24.38D14
►On 112 points
Generators in S112
(2 79)(4 81)(6 83)(8 71)(10 73)(12 75)(14 77)(16 101)(18 103)(20 105)(22 107)(24 109)(26 111)(28 99)(30 67)(32 69)(34 57)(36 59)(38 61)(40 63)(42 65)(44 93)(46 95)(48 97)(50 85)(52 87)(54 89)(56 91)
(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 99)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 71)(68 72)(69 73)(70 74)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 99)(92 100)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 99)(29 66)(30 67)(31 68)(32 69)(33 70)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)
(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 52 37 24)(2 51 38 23)(3 50 39 22)(4 49 40 21)(5 48 41 20)(6 47 42 19)(7 46 29 18)(8 45 30 17)(9 44 31 16)(10 43 32 15)(11 56 33 28)(12 55 34 27)(13 54 35 26)(14 53 36 25)(57 112 75 90)(58 111 76 89)(59 110 77 88)(60 109 78 87)(61 108 79 86)(62 107 80 85)(63 106 81 98)(64 105 82 97)(65 104 83 96)(66 103 84 95)(67 102 71 94)(68 101 72 93)(69 100 73 92)(70 99 74 91)
G:=sub<Sym(112)| (2,79)(4,81)(6,83)(8,71)(10,73)(12,75)(14,77)(16,101)(18,103)(20,105)(22,107)(24,109)(26,111)(28,99)(30,67)(32,69)(34,57)(36,59)(38,61)(40,63)(42,65)(44,93)(46,95)(48,97)(50,85)(52,87)(54,89)(56,91), (15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,99)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,71)(68,72)(69,73)(70,74)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,99)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (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), (1,52,37,24)(2,51,38,23)(3,50,39,22)(4,49,40,21)(5,48,41,20)(6,47,42,19)(7,46,29,18)(8,45,30,17)(9,44,31,16)(10,43,32,15)(11,56,33,28)(12,55,34,27)(13,54,35,26)(14,53,36,25)(57,112,75,90)(58,111,76,89)(59,110,77,88)(60,109,78,87)(61,108,79,86)(62,107,80,85)(63,106,81,98)(64,105,82,97)(65,104,83,96)(66,103,84,95)(67,102,71,94)(68,101,72,93)(69,100,73,92)(70,99,74,91)>;
G:=Group( (2,79)(4,81)(6,83)(8,71)(10,73)(12,75)(14,77)(16,101)(18,103)(20,105)(22,107)(24,109)(26,111)(28,99)(30,67)(32,69)(34,57)(36,59)(38,61)(40,63)(42,65)(44,93)(46,95)(48,97)(50,85)(52,87)(54,89)(56,91), (15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,99)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,71)(68,72)(69,73)(70,74)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,99)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (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), (1,52,37,24)(2,51,38,23)(3,50,39,22)(4,49,40,21)(5,48,41,20)(6,47,42,19)(7,46,29,18)(8,45,30,17)(9,44,31,16)(10,43,32,15)(11,56,33,28)(12,55,34,27)(13,54,35,26)(14,53,36,25)(57,112,75,90)(58,111,76,89)(59,110,77,88)(60,109,78,87)(61,108,79,86)(62,107,80,85)(63,106,81,98)(64,105,82,97)(65,104,83,96)(66,103,84,95)(67,102,71,94)(68,101,72,93)(69,100,73,92)(70,99,74,91) );
G=PermutationGroup([[(2,79),(4,81),(6,83),(8,71),(10,73),(12,75),(14,77),(16,101),(18,103),(20,105),(22,107),(24,109),(26,111),(28,99),(30,67),(32,69),(34,57),(36,59),(38,61),(40,63),(42,65),(44,93),(46,95),(48,97),(50,85),(52,87),(54,89),(56,91)], [(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,99),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,71),(68,72),(69,73),(70,74),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,99),(92,100),(93,101),(94,102),(95,103),(96,104),(97,105),(98,106)], [(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,99),(29,66),(30,67),(31,68),(32,69),(33,70),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91)], [(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)], [(1,52,37,24),(2,51,38,23),(3,50,39,22),(4,49,40,21),(5,48,41,20),(6,47,42,19),(7,46,29,18),(8,45,30,17),(9,44,31,16),(10,43,32,15),(11,56,33,28),(12,55,34,27),(13,54,35,26),(14,53,36,25),(57,112,75,90),(58,111,76,89),(59,110,77,88),(60,109,78,87),(61,108,79,86),(62,107,80,85),(63,106,81,98),(64,105,82,97),(65,104,83,96),(66,103,84,95),(67,102,71,94),(68,101,72,93),(69,100,73,92),(70,99,74,91)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 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 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D7 | D14 | Dic7 | D14 | D14 | 2+ 1+4 | D4⋊6D14 |
| kernel | C24.38D14 | C23.21D14 | D4×Dic7 | C2×C23.D7 | D4×C2×C14 | D4×C14 | C22×D4 | C22×C4 | C2×D4 | C2×D4 | C24 | C14 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 16 | 3 | 3 | 24 | 12 | 6 | 2 | 12 |
Matrix representation of C24.38D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 28 | 0 |
| 0 | 0 | 12 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 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 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 27 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 1 |
| 0 | 0 | 0 | 17 | 1 | 0 |
| 24 | 12 | 0 | 0 | 0 | 0 |
| 22 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 0 | 0 | 28 | 1 | 0 | 17 |
| 0 | 0 | 28 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,1,0,12,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,12,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[6,27,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,27,28,17,17,0,0,0,0,0,1,0,0,0,0,1,0],[24,22,0,0,0,0,12,5,0,0,0,0,0,0,12,0,28,28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,27,28,17,17] >;
C24.38D14 in GAP, Magma, Sage, TeX
C_2^4._{38}D_{14} % in TeX
G:=Group("C2^4.38D14"); // GroupNames label
G:=SmallGroup(448,1251);
// by ID
G=gap.SmallGroup(448,1251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,387,1123,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations