metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.23D14, D14⋊C4⋊2C2, (C22×C4)⋊3D7, Dic7⋊C4⋊3C2, (C22×C28)⋊2C2, (C2×C14).37D4, C14.42(C2×D4), (C2×C4).65D14, C23.D7⋊6C2, C14.18(C4○D4), C2.18(C4○D28), (C2×C14).47C23, (C2×C28).78C22, C22.9(C7⋊D4), C7⋊4(C22.D4), (C22×D7).9C22, C22.55(C22×D7), (C22×C14).39C22, (C2×Dic7).15C22, C2.6(C2×C7⋊D4), (C2×C7⋊D4).6C2, SmallGroup(224,124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.23D14
G = < a,b,c,d,e | a2=b2=c2=1, d14=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd13 >
Subgroups: 302 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C22.D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, Dic7⋊C4, D14⋊C4, C23.D7, C2×C7⋊D4, C22×C28, C23.23D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C7⋊D4, C22×D7, C4○D28, C2×C7⋊D4, C23.23D14
►On 112 points
Generators in S112
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(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)(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)
(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 14 38 51)(2 50 39 13)(3 12 40 49)(4 48 41 11)(5 10 42 47)(6 46 43 9)(7 8 44 45)(15 28 52 37)(16 36 53 27)(17 26 54 35)(18 34 55 25)(19 24 56 33)(20 32 29 23)(21 22 30 31)(57 96 111 70)(58 69 112 95)(59 94 85 68)(60 67 86 93)(61 92 87 66)(62 65 88 91)(63 90 89 64)(71 110 97 84)(72 83 98 109)(73 108 99 82)(74 81 100 107)(75 106 101 80)(76 79 102 105)(77 104 103 78)
G:=sub<Sym(112)| (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94), (1,52)(2,53)(3,54)(4,55)(5,56)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(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), (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), (1,14,38,51)(2,50,39,13)(3,12,40,49)(4,48,41,11)(5,10,42,47)(6,46,43,9)(7,8,44,45)(15,28,52,37)(16,36,53,27)(17,26,54,35)(18,34,55,25)(19,24,56,33)(20,32,29,23)(21,22,30,31)(57,96,111,70)(58,69,112,95)(59,94,85,68)(60,67,86,93)(61,92,87,66)(62,65,88,91)(63,90,89,64)(71,110,97,84)(72,83,98,109)(73,108,99,82)(74,81,100,107)(75,106,101,80)(76,79,102,105)(77,104,103,78)>;
G:=Group( (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94), (1,52)(2,53)(3,54)(4,55)(5,56)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(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), (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), (1,14,38,51)(2,50,39,13)(3,12,40,49)(4,48,41,11)(5,10,42,47)(6,46,43,9)(7,8,44,45)(15,28,52,37)(16,36,53,27)(17,26,54,35)(18,34,55,25)(19,24,56,33)(20,32,29,23)(21,22,30,31)(57,96,111,70)(58,69,112,95)(59,94,85,68)(60,67,86,93)(61,92,87,66)(62,65,88,91)(63,90,89,64)(71,110,97,84)(72,83,98,109)(73,108,99,82)(74,81,100,107)(75,106,101,80)(76,79,102,105)(77,104,103,78) );
G=PermutationGroup([[(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,95),(30,96),(31,97),(32,98),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,105),(40,106),(41,107),(42,108),(43,109),(44,110),(45,111),(46,112),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(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),(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)], [(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)], [(1,14,38,51),(2,50,39,13),(3,12,40,49),(4,48,41,11),(5,10,42,47),(6,46,43,9),(7,8,44,45),(15,28,52,37),(16,36,53,27),(17,26,54,35),(18,34,55,25),(19,24,56,33),(20,32,29,23),(21,22,30,31),(57,96,111,70),(58,69,112,95),(59,94,85,68),(60,67,86,93),(61,92,87,66),(62,65,88,91),(63,90,89,64),(71,110,97,84),(72,83,98,109),(73,108,99,82),(74,81,100,107),(75,106,101,80),(76,79,102,105),(77,104,103,78)]])
C23.23D14 is a maximal subgroup of
C22⋊C4⋊D14 C42.277D14 C24.27D14 C24.31D14 C14.2- 1+4 C14.52- 1+4 C14.62- 1+4 C42⋊10D14 C42.96D14 C42.104D14 C42⋊16D14 C42.113D14 C42.114D14 C42⋊17D14 C42.115D14 C42.116D14 C42.118D14 C14.422+ 1+4 C14.442+ 1+4 C14.482+ 1+4 C14.492+ 1+4 C14.202- 1+4 C14.222- 1+4 C14.582+ 1+4 C14.262- 1+4 C14.792- 1+4 C4⋊C4.197D14 D7×C22.D4 C14.1202+ 1+4 C4⋊C4⋊28D14 C14.852- 1+4 C24.72D14 C24⋊7D14 C14.442- 1+4 C14.1042- 1+4 C14.1452+ 1+4
C23.23D14 is a maximal quotient of
(C2×C42).D7 (C2×C42)⋊D7 C24.9D14 C24.14D14 (C2×C14).40D8 C4⋊C4.228D14 C4⋊C4.230D14 C4⋊C4.231D14 (C2×C28).287D4 (C2×C28).288D4 (C2×C28).289D4 (C2×C28).290D4 C4⋊C4.233D14 C4⋊C4.236D14 C24.62D14 C24.63D14 C23.28D28
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C4○D28 |
| kernel | C23.23D14 | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | C22×C28 | C2×C14 | C22×C4 | C14 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 4 | 6 | 3 | 12 | 24 |
Matrix representation of C23.23D14 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 24 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 1 | 0 | 0 |
| 28 | 2 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 2 | 17 |
| 1 | 28 | 0 | 0 |
| 2 | 28 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 2 | 17 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,24,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,28,0,0,1,2,0,0,0,0,12,2,0,0,0,17],[1,2,0,0,28,28,0,0,0,0,12,2,0,0,1,17] >;
C23.23D14 in GAP, Magma, Sage, TeX
C_2^3._{23}D_{14} % in TeX
G:=Group("C2^3.23D14"); // GroupNames label
G:=SmallGroup(224,124);
// by ID
G=gap.SmallGroup(224,124);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,218,86,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^14=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^13>;
// generators/relations