metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.8D28, C24.1D14, C23.1Dic14, (C22×C28)⋊1C4, C23.D7⋊5C4, C23.8(C4×D7), (C22×C4)⋊1Dic7, (C22×C14).6Q8, (C2×C14).17C42, (C22×C14).41D4, C7⋊1(C23.9D4), C22.8(C4×Dic7), C14.17(C23⋊C4), C23.46(C7⋊D4), C22.8(C4⋊Dic7), C2.2(C23⋊Dic7), C23.21(C2×Dic7), C22.16(D14⋊C4), C22.1(Dic7⋊C4), (C23×C14).22C22, C14.3(C2.C42), C2.4(C14.C42), C22.24(C23.D7), (C2×C14).1(C4⋊C4), (C2×C22⋊C4).2D7, (C14×C22⋊C4).1C2, (C2×C23.D7).1C2, (C2×C14).9(C22⋊C4), (C22×C14).28(C2×C4), SmallGroup(448,83)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.8D28
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >
Subgroups: 628 in 142 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22 [×3], C22 [×4], C22 [×10], C7, C2×C4 [×12], C23 [×3], C23 [×4], C23 [×2], C14, C14 [×2], C14 [×6], C22⋊C4 [×8], C22×C4 [×2], C22×C4 [×2], C24, Dic7 [×4], C28 [×2], C2×C14 [×3], C2×C14 [×4], C2×C14 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×Dic7 [×8], C2×C28 [×4], C22×C14 [×3], C22×C14 [×4], C22×C14 [×2], C23.9D4, C23.D7 [×4], C23.D7 [×2], C7×C22⋊C4 [×2], C22×Dic7 [×2], C22×C28 [×2], C23×C14, C2×C23.D7 [×2], C14×C22⋊C4, C23.8D28
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D7, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic7 [×2], D14, C2.C42, C23⋊C4 [×2], Dic14, C4×D7 [×2], D28, C2×Dic7, C7⋊D4 [×2], C23.9D4, C4×Dic7, Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4 [×2], C23.D7, C14.C42, C23⋊Dic7 [×2], C23.8D28
►On 112 points
Generators in S112
(1 45)(2 59)(3 47)(4 61)(5 49)(6 63)(7 51)(8 65)(9 53)(10 67)(11 55)(12 69)(13 29)(14 71)(15 31)(16 73)(17 33)(18 75)(19 35)(20 77)(21 37)(22 79)(23 39)(24 81)(25 41)(26 83)(27 43)(28 57)(30 92)(32 94)(34 96)(36 98)(38 100)(40 102)(42 104)(44 106)(46 108)(48 110)(50 112)(52 86)(54 88)(56 90)(58 107)(60 109)(62 111)(64 85)(66 87)(68 89)(70 91)(72 93)(74 95)(76 97)(78 99)(80 101)(82 103)(84 105)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)
(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 51 107 64)(2 6)(3 49 109 62)(5 47 111 60)(7 45 85 58)(8 28)(9 43 87 84)(10 26)(11 41 89 82)(12 24)(13 39 91 80)(14 22)(15 37 93 78)(16 20)(17 35 95 76)(19 33 97 74)(21 31 99 72)(23 29 101 70)(25 55 103 68)(27 53 105 66)(30 79)(32 77)(34 75)(36 73)(38 71)(40 69)(42 67)(44 65)(46 63)(48 61)(50 59)(52 57)(54 83)(56 81)(86 106)(88 104)(90 102)(92 100)(94 98)(108 112)
G:=sub<Sym(112)| (1,45)(2,59)(3,47)(4,61)(5,49)(6,63)(7,51)(8,65)(9,53)(10,67)(11,55)(12,69)(13,29)(14,71)(15,31)(16,73)(17,33)(18,75)(19,35)(20,77)(21,37)(22,79)(23,39)(24,81)(25,41)(26,83)(27,43)(28,57)(30,92)(32,94)(34,96)(36,98)(38,100)(40,102)(42,104)(44,106)(46,108)(48,110)(50,112)(52,86)(54,88)(56,90)(58,107)(60,109)(62,111)(64,85)(66,87)(68,89)(70,91)(72,93)(74,95)(76,97)(78,99)(80,101)(82,103)(84,105), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69), (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), (1,51,107,64)(2,6)(3,49,109,62)(5,47,111,60)(7,45,85,58)(8,28)(9,43,87,84)(10,26)(11,41,89,82)(12,24)(13,39,91,80)(14,22)(15,37,93,78)(16,20)(17,35,95,76)(19,33,97,74)(21,31,99,72)(23,29,101,70)(25,55,103,68)(27,53,105,66)(30,79)(32,77)(34,75)(36,73)(38,71)(40,69)(42,67)(44,65)(46,63)(48,61)(50,59)(52,57)(54,83)(56,81)(86,106)(88,104)(90,102)(92,100)(94,98)(108,112)>;
G:=Group( (1,45)(2,59)(3,47)(4,61)(5,49)(6,63)(7,51)(8,65)(9,53)(10,67)(11,55)(12,69)(13,29)(14,71)(15,31)(16,73)(17,33)(18,75)(19,35)(20,77)(21,37)(22,79)(23,39)(24,81)(25,41)(26,83)(27,43)(28,57)(30,92)(32,94)(34,96)(36,98)(38,100)(40,102)(42,104)(44,106)(46,108)(48,110)(50,112)(52,86)(54,88)(56,90)(58,107)(60,109)(62,111)(64,85)(66,87)(68,89)(70,91)(72,93)(74,95)(76,97)(78,99)(80,101)(82,103)(84,105), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69), (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), (1,51,107,64)(2,6)(3,49,109,62)(5,47,111,60)(7,45,85,58)(8,28)(9,43,87,84)(10,26)(11,41,89,82)(12,24)(13,39,91,80)(14,22)(15,37,93,78)(16,20)(17,35,95,76)(19,33,97,74)(21,31,99,72)(23,29,101,70)(25,55,103,68)(27,53,105,66)(30,79)(32,77)(34,75)(36,73)(38,71)(40,69)(42,67)(44,65)(46,63)(48,61)(50,59)(52,57)(54,83)(56,81)(86,106)(88,104)(90,102)(92,100)(94,98)(108,112) );
G=PermutationGroup([(1,45),(2,59),(3,47),(4,61),(5,49),(6,63),(7,51),(8,65),(9,53),(10,67),(11,55),(12,69),(13,29),(14,71),(15,31),(16,73),(17,33),(18,75),(19,35),(20,77),(21,37),(22,79),(23,39),(24,81),(25,41),(26,83),(27,43),(28,57),(30,92),(32,94),(34,96),(36,98),(38,100),(40,102),(42,104),(44,106),(46,108),(48,110),(50,112),(52,86),(54,88),(56,90),(58,107),(60,109),(62,111),(64,85),(66,87),(68,89),(70,91),(72,93),(74,95),(76,97),(78,99),(80,101),(82,103),(84,105)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,107),(2,108),(3,109),(4,110),(5,111),(6,112),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,57),(45,58),(46,59),(47,60),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69)], [(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)], [(1,51,107,64),(2,6),(3,49,109,62),(5,47,111,60),(7,45,85,58),(8,28),(9,43,87,84),(10,26),(11,41,89,82),(12,24),(13,39,91,80),(14,22),(15,37,93,78),(16,20),(17,35,95,76),(19,33,97,74),(21,31,99,72),(23,29,101,70),(25,55,103,68),(27,53,105,66),(30,79),(32,77),(34,75),(36,73),(38,71),(40,69),(42,67),(44,65),(46,63),(48,61),(50,59),(52,57),(54,83),(56,81),(86,106),(88,104),(90,102),(92,100),(94,98),(108,112)])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| 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 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D7 | Dic7 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C23⋊C4 | C23⋊Dic7 |
| kernel | C23.8D28 | C2×C23.D7 | C14×C22⋊C4 | C23.D7 | C22×C28 | C22×C14 | C22×C14 | C2×C22⋊C4 | C22×C4 | C24 | C23 | C23 | C23 | C23 | C14 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 3 | 6 | 3 | 6 | 12 | 6 | 12 | 2 | 12 |
Matrix representation of C23.8D28 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 14 |
| 0 | 0 | 0 | 0 | 15 | 20 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 15 |
| 0 | 0 | 0 | 0 | 14 | 9 |
| 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 |
| 12 | 16 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 22 |
| 0 | 0 | 0 | 0 | 7 | 28 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 7 | 28 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 11 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 10 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 22 |
| 0 | 0 | 0 | 0 | 10 | 10 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,9,15,0,0,0,0,14,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,20,14,0,0,0,0,15,9],[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],[12,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,19,7,0,0,0,0,22,28,0,0,19,7,0,0,0,0,22,28,0,0],[12,11,0,0,0,0,0,17,0,0,0,0,0,0,8,8,0,0,0,0,10,21,0,0,0,0,0,0,19,10,0,0,0,0,22,10] >;
C23.8D28 in GAP, Magma, Sage, TeX
C_2^3._8D_{28} % in TeX
G:=Group("C2^3.8D28"); // GroupNames label
G:=SmallGroup(448,83);
// by ID
G=gap.SmallGroup(448,83);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=a*b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;
// generators/relations