metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.6Dic14, M4(2).30D14, C28.37(C4⋊C4), (C2×C28).28Q8, (C2×C28).483D4, C28.441(C2×D4), C4.Dic7.6C4, C28.68(C22×C4), (C2×C4).16Dic14, C28.53D4⋊13C2, (C22×C14).16Q8, C7⋊2(M4(2).C4), C4.21(Dic7⋊C4), (C2×C28).415C23, (C22×C4).134D14, (C2×M4(2)).16D7, C22.4(C2×Dic14), (C14×M4(2)).27C2, C4.Dic7.41C22, C22.17(Dic7⋊C4), (C22×C28).183C22, (C7×M4(2)).33C22, C7⋊C8.5(C2×C4), C4.90(C2×C4×D7), C14.52(C2×C4⋊C4), (C2×C4).49(C4×D7), C4.131(C2×C7⋊D4), (C2×C14).11(C2×Q8), (C2×C14).54(C4⋊C4), (C2×C28).103(C2×C4), (C2×C7⋊C8).143C22, C2.19(C2×Dic7⋊C4), (C2×C4).194(C7⋊D4), (C2×C4).511(C22×D7), (C2×C4.Dic7).24C2, SmallGroup(448,658)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.Dic14
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=bd14, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd27 >
Subgroups: 292 in 102 conjugacy classes, 59 normal (39 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C23, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C28, C2×C14, C2×C14, C8.C4, C2×M4(2), C2×M4(2), C7⋊C8, C7⋊C8, C56, C2×C28, C22×C14, M4(2).C4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C28.53D4, C2×C4.Dic7, C14×M4(2), C23.Dic14
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, Dic14, C4×D7, C7⋊D4, C22×D7, M4(2).C4, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C2×Dic7⋊C4, C23.Dic14
►On 112 points
Generators in S112
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 42)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(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 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(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 85 15 99 29 57 43 71)(2 112 44 98 30 84 16 70)(3 111 17 69 31 83 45 97)(4 82 46 68 32 110 18 96)(5 81 19 95 33 109 47 67)(6 108 48 94 34 80 20 66)(7 107 21 65 35 79 49 93)(8 78 50 64 36 106 22 92)(9 77 23 91 37 105 51 63)(10 104 52 90 38 76 24 62)(11 103 25 61 39 75 53 89)(12 74 54 60 40 102 26 88)(13 73 27 87 41 101 55 59)(14 100 56 86 42 72 28 58)
G:=sub<Sym(112)| (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(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)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,85,15,99,29,57,43,71)(2,112,44,98,30,84,16,70)(3,111,17,69,31,83,45,97)(4,82,46,68,32,110,18,96)(5,81,19,95,33,109,47,67)(6,108,48,94,34,80,20,66)(7,107,21,65,35,79,49,93)(8,78,50,64,36,106,22,92)(9,77,23,91,37,105,51,63)(10,104,52,90,38,76,24,62)(11,103,25,61,39,75,53,89)(12,74,54,60,40,102,26,88)(13,73,27,87,41,101,55,59)(14,100,56,86,42,72,28,58)>;
G:=Group( (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(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)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,85,15,99,29,57,43,71)(2,112,44,98,30,84,16,70)(3,111,17,69,31,83,45,97)(4,82,46,68,32,110,18,96)(5,81,19,95,33,109,47,67)(6,108,48,94,34,80,20,66)(7,107,21,65,35,79,49,93)(8,78,50,64,36,106,22,92)(9,77,23,91,37,105,51,63)(10,104,52,90,38,76,24,62)(11,103,25,61,39,75,53,89)(12,74,54,60,40,102,26,88)(13,73,27,87,41,101,55,59)(14,100,56,86,42,72,28,58) );
G=PermutationGroup([[(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(2,30),(4,32),(6,34),(8,36),(10,38),(12,40),(14,42),(16,44),(18,46),(20,48),(22,50),(24,52),(26,54),(28,56),(58,86),(60,88),(62,90),(64,92),(66,94),(68,96),(70,98),(72,100),(74,102),(76,104),(78,106),(80,108),(82,110),(84,112)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(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),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(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)], [(1,85,15,99,29,57,43,71),(2,112,44,98,30,84,16,70),(3,111,17,69,31,83,45,97),(4,82,46,68,32,110,18,96),(5,81,19,95,33,109,47,67),(6,108,48,94,34,80,20,66),(7,107,21,65,35,79,49,93),(8,78,50,64,36,106,22,92),(9,77,23,91,37,105,51,63),(10,104,52,90,38,76,24,62),(11,103,25,61,39,75,53,89),(12,74,54,60,40,102,26,88),(13,73,27,87,41,101,55,59),(14,100,56,86,42,72,28,58)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D7 | D14 | D14 | Dic14 | C4×D7 | C7⋊D4 | Dic14 | M4(2).C4 | C23.Dic14 |
| kernel | C23.Dic14 | C28.53D4 | C2×C4.Dic7 | C14×M4(2) | C4.Dic7 | C2×C28 | C2×C28 | C22×C14 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 1 | 1 | 3 | 6 | 3 | 6 | 12 | 12 | 6 | 2 | 12 |
Matrix representation of C23.Dic14 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 61 | 0 | 112 | 0 |
| 65 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 28 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 65 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 2 | 16 | 0 | 0 |
| 109 | 111 | 0 | 0 |
| 49 | 36 | 0 | 56 |
| 34 | 68 | 64 | 0 |
| 44 | 0 | 106 | 0 |
| 75 | 0 | 15 | 1 |
| 85 | 0 | 69 | 0 |
| 90 | 15 | 55 | 0 |
G:=sub<GL(4,GF(113))| [1,0,61,65,0,1,0,0,0,0,112,0,0,0,0,112],[1,28,0,65,0,112,0,0,0,0,1,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[2,109,49,34,16,111,36,68,0,0,0,64,0,0,56,0],[44,75,85,90,0,0,0,15,106,15,69,55,0,1,0,0] >;
C23.Dic14 in GAP, Magma, Sage, TeX
C_2^3.{\rm Dic}_{14} % in TeX
G:=Group("C2^3.Dic14"); // GroupNames label
G:=SmallGroup(448,658);
// by ID
G=gap.SmallGroup(448,658);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,422,58,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=b*d^14,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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*c*d^27>;
// generators/relations