metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.11D14, Dic7⋊C4⋊7C2, (C2×Dic7)⋊3C4, (C4×Dic7)⋊9C2, (C2×C4).26D14, C22⋊C4.3D7, C22.6(C4×D7), C7⋊2(C42⋊C2), C14.5(C22×C4), Dic7.8(C2×C4), C23.D7.1C2, C14.20(C4○D4), C2.1(D4⋊2D7), (C2×C14).18C23, (C2×C28).50C22, (C22×C14).7C22, (C22×Dic7).2C2, C22.12(C22×D7), (C2×Dic7).46C22, C2.7(C2×C4×D7), (C2×C14).4(C2×C4), (C7×C22⋊C4).3C2, SmallGroup(224,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.11D14
G = < a,b,c,d,e | a2=b2=c2=1, d14=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d13 >
Subgroups: 230 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C2×Dic7, C2×Dic7, C2×C28, C22×C14, C4×Dic7, Dic7⋊C4, C23.D7, C7×C22⋊C4, C22×Dic7, C23.11D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, D14, C42⋊C2, C4×D7, C22×D7, C2×C4×D7, D4⋊2D7, C23.11D14
►On 112 points
Generators in S112
(2 37)(4 39)(6 41)(8 43)(10 45)(12 47)(14 49)(16 51)(18 53)(20 55)(22 29)(24 31)(26 33)(28 35)(58 98)(60 100)(62 102)(64 104)(66 106)(68 108)(70 110)(72 112)(74 86)(76 88)(78 90)(80 92)(82 94)(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 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(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 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)
(1 110 50 84)(2 95 51 69)(3 108 52 82)(4 93 53 67)(5 106 54 80)(6 91 55 65)(7 104 56 78)(8 89 29 63)(9 102 30 76)(10 87 31 61)(11 100 32 74)(12 85 33 59)(13 98 34 72)(14 111 35 57)(15 96 36 70)(16 109 37 83)(17 94 38 68)(18 107 39 81)(19 92 40 66)(20 105 41 79)(21 90 42 64)(22 103 43 77)(23 88 44 62)(24 101 45 75)(25 86 46 60)(26 99 47 73)(27 112 48 58)(28 97 49 71)
G:=sub<Sym(112)| (2,37)(4,39)(6,41)(8,43)(10,45)(12,47)(14,49)(16,51)(18,53)(20,55)(22,29)(24,31)(26,33)(28,35)(58,98)(60,100)(62,102)(64,104)(66,106)(68,108)(70,110)(72,112)(74,86)(76,88)(78,90)(80,92)(82,94)(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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,110,50,84)(2,95,51,69)(3,108,52,82)(4,93,53,67)(5,106,54,80)(6,91,55,65)(7,104,56,78)(8,89,29,63)(9,102,30,76)(10,87,31,61)(11,100,32,74)(12,85,33,59)(13,98,34,72)(14,111,35,57)(15,96,36,70)(16,109,37,83)(17,94,38,68)(18,107,39,81)(19,92,40,66)(20,105,41,79)(21,90,42,64)(22,103,43,77)(23,88,44,62)(24,101,45,75)(25,86,46,60)(26,99,47,73)(27,112,48,58)(28,97,49,71)>;
G:=Group( (2,37)(4,39)(6,41)(8,43)(10,45)(12,47)(14,49)(16,51)(18,53)(20,55)(22,29)(24,31)(26,33)(28,35)(58,98)(60,100)(62,102)(64,104)(66,106)(68,108)(70,110)(72,112)(74,86)(76,88)(78,90)(80,92)(82,94)(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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,110,50,84)(2,95,51,69)(3,108,52,82)(4,93,53,67)(5,106,54,80)(6,91,55,65)(7,104,56,78)(8,89,29,63)(9,102,30,76)(10,87,31,61)(11,100,32,74)(12,85,33,59)(13,98,34,72)(14,111,35,57)(15,96,36,70)(16,109,37,83)(17,94,38,68)(18,107,39,81)(19,92,40,66)(20,105,41,79)(21,90,42,64)(22,103,43,77)(23,88,44,62)(24,101,45,75)(25,86,46,60)(26,99,47,73)(27,112,48,58)(28,97,49,71) );
G=PermutationGroup([[(2,37),(4,39),(6,41),(8,43),(10,45),(12,47),(14,49),(16,51),(18,53),(20,55),(22,29),(24,31),(26,33),(28,35),(58,98),(60,100),(62,102),(64,104),(66,106),(68,108),(70,110),(72,112),(74,86),(76,88),(78,90),(80,92),(82,94),(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)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(28,35),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(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)], [(1,110,50,84),(2,95,51,69),(3,108,52,82),(4,93,53,67),(5,106,54,80),(6,91,55,65),(7,104,56,78),(8,89,29,63),(9,102,30,76),(10,87,31,61),(11,100,32,74),(12,85,33,59),(13,98,34,72),(14,111,35,57),(15,96,36,70),(16,109,37,83),(17,94,38,68),(18,107,39,81),(19,92,40,66),(20,105,41,79),(21,90,42,64),(22,103,43,77),(23,88,44,62),(24,101,45,75),(25,86,46,60),(26,99,47,73),(27,112,48,58),(28,97,49,71)]])
C23.11D14 is a maximal subgroup of
C23⋊C4⋊5D7 C24.24D14 C24.31D14 C42.87D14 D7×C42⋊C2 C42.96D14 C4×D4⋊2D7 C42.105D14 C42.108D14 C42⋊16D14 C24.56D14 C24.32D14 C24.34D14 C4⋊C4.178D14 C14.342+ 1+4 C14.442+ 1+4 C14.452+ 1+4 (Q8×Dic7)⋊C2 C14.752- 1+4 C14.532+ 1+4 C14.772- 1+4 C14.792- 1+4 C4⋊C4.197D14 C14.802- 1+4 C4⋊C4⋊28D14 C14.642+ 1+4 C14.842- 1+4 C42.137D14 C42.138D14 C42.139D14 C42.234D14 C42.159D14 C42.160D14 C42.189D14 C42.162D14
C23.11D14 is a maximal quotient of
Dic7.5C42 Dic7⋊C42 C7⋊(C42⋊8C4) C7⋊(C42⋊5C4) C4⋊Dic7⋊7C4 C4⋊Dic7⋊8C4 Dic7.5M4(2) Dic7.M4(2) C56⋊C4⋊C2 C22⋊C4×Dic7 C24.44D14 C23.42D28 C24.3D14 C24.4D14
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 7 | 7 | 7 | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D4⋊2D7 |
| kernel | C23.11D14 | C4×Dic7 | Dic7⋊C4 | C23.D7 | C7×C22⋊C4 | C22×Dic7 | C2×Dic7 | C22⋊C4 | C14 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 4 | 6 | 3 | 12 | 6 |
Matrix representation of C23.11D14 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 2 | 10 | 0 | 0 |
| 2 | 24 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 21 | 2 | 0 | 0 |
| 11 | 8 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 17 | 0 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[2,2,0,0,10,24,0,0,0,0,0,1,0,0,1,0],[21,11,0,0,2,8,0,0,0,0,0,17,0,0,17,0] >;
C23.11D14 in GAP, Magma, Sage, TeX
C_2^3._{11}D_{14} % in TeX
G:=Group("C2^3.11D14"); // GroupNames label
G:=SmallGroup(224,72);
// by ID
G=gap.SmallGroup(224,72);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,188,50,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^14=b,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^13>;
// generators/relations