direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22⋊Q8, C28.62D4, C4⋊C4⋊3C14, C22⋊(C7×Q8), (C2×C14)⋊2Q8, (C2×Q8)⋊1C14, (Q8×C14)⋊8C2, C2.6(D4×C14), C4.13(C7×D4), C2.3(Q8×C14), C14.69(C2×D4), C14.20(C2×Q8), C22⋊C4.1C14, (C22×C4).5C14, C23.9(C2×C14), C14.42(C4○D4), (C22×C28).15C2, (C2×C14).77C23, (C2×C28).124C22, C22.12(C22×C14), (C22×C14).28C22, (C7×C4⋊C4)⋊12C2, C2.5(C7×C4○D4), (C2×C4).4(C2×C14), (C7×C22⋊C4).4C2, SmallGroup(224,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22⋊Q8
G = < a,b,c,d,e | a7=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 100 in 74 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊Q8, C2×C28, C2×C28, C2×C28, C7×Q8, C22×C14, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, Q8×C14, C7×C22⋊Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C2×C14, C22⋊Q8, C7×D4, C7×Q8, C22×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊Q8
►On 112 points
Generators in S112
(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)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 50 38 74)(2 51 39 75)(3 52 40 76)(4 53 41 77)(5 54 42 71)(6 55 36 72)(7 56 37 73)(8 90 27 98)(9 91 28 92)(10 85 22 93)(11 86 23 94)(12 87 24 95)(13 88 25 96)(14 89 26 97)(15 81 109 105)(16 82 110 99)(17 83 111 100)(18 84 112 101)(19 78 106 102)(20 79 107 103)(21 80 108 104)(29 65 44 57)(30 66 45 58)(31 67 46 59)(32 68 47 60)(33 69 48 61)(34 70 49 62)(35 64 43 63)
(1 78 38 102)(2 79 39 103)(3 80 40 104)(4 81 41 105)(5 82 42 99)(6 83 36 100)(7 84 37 101)(8 70 27 62)(9 64 28 63)(10 65 22 57)(11 66 23 58)(12 67 24 59)(13 68 25 60)(14 69 26 61)(15 77 109 53)(16 71 110 54)(17 72 111 55)(18 73 112 56)(19 74 106 50)(20 75 107 51)(21 76 108 52)(29 93 44 85)(30 94 45 86)(31 95 46 87)(32 96 47 88)(33 97 48 89)(34 98 49 90)(35 92 43 91)
G:=sub<Sym(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), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,90,27,98)(9,91,28,92)(10,85,22,93)(11,86,23,94)(12,87,24,95)(13,88,25,96)(14,89,26,97)(15,81,109,105)(16,82,110,99)(17,83,111,100)(18,84,112,101)(19,78,106,102)(20,79,107,103)(21,80,108,104)(29,65,44,57)(30,66,45,58)(31,67,46,59)(32,68,47,60)(33,69,48,61)(34,70,49,62)(35,64,43,63), (1,78,38,102)(2,79,39,103)(3,80,40,104)(4,81,41,105)(5,82,42,99)(6,83,36,100)(7,84,37,101)(8,70,27,62)(9,64,28,63)(10,65,22,57)(11,66,23,58)(12,67,24,59)(13,68,25,60)(14,69,26,61)(15,77,109,53)(16,71,110,54)(17,72,111,55)(18,73,112,56)(19,74,106,50)(20,75,107,51)(21,76,108,52)(29,93,44,85)(30,94,45,86)(31,95,46,87)(32,96,47,88)(33,97,48,89)(34,98,49,90)(35,92,43,91)>;
G:=Group( (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), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,90,27,98)(9,91,28,92)(10,85,22,93)(11,86,23,94)(12,87,24,95)(13,88,25,96)(14,89,26,97)(15,81,109,105)(16,82,110,99)(17,83,111,100)(18,84,112,101)(19,78,106,102)(20,79,107,103)(21,80,108,104)(29,65,44,57)(30,66,45,58)(31,67,46,59)(32,68,47,60)(33,69,48,61)(34,70,49,62)(35,64,43,63), (1,78,38,102)(2,79,39,103)(3,80,40,104)(4,81,41,105)(5,82,42,99)(6,83,36,100)(7,84,37,101)(8,70,27,62)(9,64,28,63)(10,65,22,57)(11,66,23,58)(12,67,24,59)(13,68,25,60)(14,69,26,61)(15,77,109,53)(16,71,110,54)(17,72,111,55)(18,73,112,56)(19,74,106,50)(20,75,107,51)(21,76,108,52)(29,93,44,85)(30,94,45,86)(31,95,46,87)(32,96,47,88)(33,97,48,89)(34,98,49,90)(35,92,43,91) );
G=PermutationGroup([[(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)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(1,50,38,74),(2,51,39,75),(3,52,40,76),(4,53,41,77),(5,54,42,71),(6,55,36,72),(7,56,37,73),(8,90,27,98),(9,91,28,92),(10,85,22,93),(11,86,23,94),(12,87,24,95),(13,88,25,96),(14,89,26,97),(15,81,109,105),(16,82,110,99),(17,83,111,100),(18,84,112,101),(19,78,106,102),(20,79,107,103),(21,80,108,104),(29,65,44,57),(30,66,45,58),(31,67,46,59),(32,68,47,60),(33,69,48,61),(34,70,49,62),(35,64,43,63)], [(1,78,38,102),(2,79,39,103),(3,80,40,104),(4,81,41,105),(5,82,42,99),(6,83,36,100),(7,84,37,101),(8,70,27,62),(9,64,28,63),(10,65,22,57),(11,66,23,58),(12,67,24,59),(13,68,25,60),(14,69,26,61),(15,77,109,53),(16,71,110,54),(17,72,111,55),(18,73,112,56),(19,74,106,50),(20,75,107,51),(21,76,108,52),(29,93,44,85),(30,94,45,86),(31,95,46,87),(32,96,47,88),(33,97,48,89),(34,98,49,90),(35,92,43,91)]])
C7×C22⋊Q8 is a maximal subgroup of
C4⋊C4⋊Dic7 C22⋊Q8.D7 (C2×C14).Q16 C14.(C4○D8) D28.36D4 D28.37D4 C7⋊C8⋊24D4 C7⋊C8⋊6D4 Dic14.37D4 C7⋊C8.29D4 C7⋊C8.6D4 (Q8×Dic7)⋊C2 C14.752- 1+4 C22⋊Q8⋊25D7 C14.152- 1+4 C4⋊C4⋊26D14 C14.162- 1+4 C14.172- 1+4 D28⋊21D4 D28⋊22D4 Dic14⋊21D4 Dic14⋊22D4 C14.512+ 1+4 C14.1182+ 1+4 C14.522+ 1+4 C14.532+ 1+4 C14.202- 1+4 C14.212- 1+4 C14.222- 1+4 C14.232- 1+4 C14.772- 1+4 C14.242- 1+4 C14.562+ 1+4 C14.572+ 1+4 C14.582+ 1+4 C14.262- 1+4 C7×D4×Q8
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | Q8 | C4○D4 | C7×D4 | C7×Q8 | C7×C4○D4 |
| kernel | C7×C22⋊Q8 | C7×C22⋊C4 | C7×C4⋊C4 | C22×C28 | Q8×C14 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C28 | C2×C14 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 6 | 12 | 18 | 6 | 6 | 2 | 2 | 2 | 12 | 12 | 12 |
Matrix representation of C7×C22⋊Q8 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[12,0,0,0,0,17,0,0,0,0,28,0,0,0,0,28],[0,28,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C7×C22⋊Q8 in GAP, Magma, Sage, TeX
C_7\times C_2^2\rtimes Q_8
% in TeX
G:=Group("C7xC2^2:Q8"); // GroupNames label
G:=SmallGroup(224,157);
// by ID
G=gap.SmallGroup(224,157);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,697,343,2090]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*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=d^-1>;
// generators/relations