direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8.C22, Q16⋊2C14, C28.64D4, SD16⋊2C14, M4(2)⋊2C14, C28.49C23, C56.13C22, C8.(C2×C14), (C2×Q8)⋊4C14, (C7×Q16)⋊6C2, C4.15(C7×D4), (Q8×C14)⋊11C2, (C7×SD16)⋊6C2, C4○D4.2C14, D4.3(C2×C14), (C2×C14).25D4, C2.16(D4×C14), C14.79(C2×D4), Q8.3(C2×C14), C22.6(C7×D4), (C7×M4(2))⋊6C2, C4.6(C22×C14), (C2×C28).70C22, (C7×D4).13C22, (C7×Q8).14C22, (C7×C4○D4).5C2, (C2×C4).11(C2×C14), SmallGroup(224,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.C22
G = < a,b,c,d | a7=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C14, C14, M4(2), SD16, Q16, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C8.C22, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C7×M4(2), C7×SD16, C7×Q16, Q8×C14, C7×C4○D4, C7×C8.C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C8.C22, C7×D4, C22×C14, D4×C14, C7×C8.C22
►On 112 points
Generators in S112
(1 87 42 28 93 79 34)(2 88 43 29 94 80 35)(3 81 44 30 95 73 36)(4 82 45 31 96 74 37)(5 83 46 32 89 75 38)(6 84 47 25 90 76 39)(7 85 48 26 91 77 40)(8 86 41 27 92 78 33)(9 111 21 52 69 103 57)(10 112 22 53 70 104 58)(11 105 23 54 71 97 59)(12 106 24 55 72 98 60)(13 107 17 56 65 99 61)(14 108 18 49 66 100 62)(15 109 19 50 67 101 63)(16 110 20 51 68 102 64)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 51)(50 54)(53 55)(58 60)(59 63)(62 64)(66 68)(67 71)(70 72)(73 77)(74 80)(76 78)(81 85)(82 88)(84 86)(90 92)(91 95)(94 96)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)
(1 97)(2 102)(3 99)(4 104)(5 101)(6 98)(7 103)(8 100)(9 48)(10 45)(11 42)(12 47)(13 44)(14 41)(15 46)(16 43)(17 95)(18 92)(19 89)(20 94)(21 91)(22 96)(23 93)(24 90)(25 106)(26 111)(27 108)(28 105)(29 110)(30 107)(31 112)(32 109)(33 66)(34 71)(35 68)(36 65)(37 70)(38 67)(39 72)(40 69)(49 78)(50 75)(51 80)(52 77)(53 74)(54 79)(55 76)(56 73)(57 85)(58 82)(59 87)(60 84)(61 81)(62 86)(63 83)(64 88)
G:=sub<Sym(112)| (1,87,42,28,93,79,34)(2,88,43,29,94,80,35)(3,81,44,30,95,73,36)(4,82,45,31,96,74,37)(5,83,46,32,89,75,38)(6,84,47,25,90,76,39)(7,85,48,26,91,77,40)(8,86,41,27,92,78,33)(9,111,21,52,69,103,57)(10,112,22,53,70,104,58)(11,105,23,54,71,97,59)(12,106,24,55,72,98,60)(13,107,17,56,65,99,61)(14,108,18,49,66,100,62)(15,109,19,50,67,101,63)(16,110,20,51,68,102,64), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86)(90,92)(91,95)(94,96)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110), (1,97)(2,102)(3,99)(4,104)(5,101)(6,98)(7,103)(8,100)(9,48)(10,45)(11,42)(12,47)(13,44)(14,41)(15,46)(16,43)(17,95)(18,92)(19,89)(20,94)(21,91)(22,96)(23,93)(24,90)(25,106)(26,111)(27,108)(28,105)(29,110)(30,107)(31,112)(32,109)(33,66)(34,71)(35,68)(36,65)(37,70)(38,67)(39,72)(40,69)(49,78)(50,75)(51,80)(52,77)(53,74)(54,79)(55,76)(56,73)(57,85)(58,82)(59,87)(60,84)(61,81)(62,86)(63,83)(64,88)>;
G:=Group( (1,87,42,28,93,79,34)(2,88,43,29,94,80,35)(3,81,44,30,95,73,36)(4,82,45,31,96,74,37)(5,83,46,32,89,75,38)(6,84,47,25,90,76,39)(7,85,48,26,91,77,40)(8,86,41,27,92,78,33)(9,111,21,52,69,103,57)(10,112,22,53,70,104,58)(11,105,23,54,71,97,59)(12,106,24,55,72,98,60)(13,107,17,56,65,99,61)(14,108,18,49,66,100,62)(15,109,19,50,67,101,63)(16,110,20,51,68,102,64), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(73,77)(74,80)(76,78)(81,85)(82,88)(84,86)(90,92)(91,95)(94,96)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110), (1,97)(2,102)(3,99)(4,104)(5,101)(6,98)(7,103)(8,100)(9,48)(10,45)(11,42)(12,47)(13,44)(14,41)(15,46)(16,43)(17,95)(18,92)(19,89)(20,94)(21,91)(22,96)(23,93)(24,90)(25,106)(26,111)(27,108)(28,105)(29,110)(30,107)(31,112)(32,109)(33,66)(34,71)(35,68)(36,65)(37,70)(38,67)(39,72)(40,69)(49,78)(50,75)(51,80)(52,77)(53,74)(54,79)(55,76)(56,73)(57,85)(58,82)(59,87)(60,84)(61,81)(62,86)(63,83)(64,88) );
G=PermutationGroup([[(1,87,42,28,93,79,34),(2,88,43,29,94,80,35),(3,81,44,30,95,73,36),(4,82,45,31,96,74,37),(5,83,46,32,89,75,38),(6,84,47,25,90,76,39),(7,85,48,26,91,77,40),(8,86,41,27,92,78,33),(9,111,21,52,69,103,57),(10,112,22,53,70,104,58),(11,105,23,54,71,97,59),(12,106,24,55,72,98,60),(13,107,17,56,65,99,61),(14,108,18,49,66,100,62),(15,109,19,50,67,101,63),(16,110,20,51,68,102,64)], [(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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24),(25,27),(26,30),(29,31),(33,39),(35,37),(36,40),(41,47),(43,45),(44,48),(49,51),(50,54),(53,55),(58,60),(59,63),(62,64),(66,68),(67,71),(70,72),(73,77),(74,80),(76,78),(81,85),(82,88),(84,86),(90,92),(91,95),(94,96),(97,101),(98,104),(100,102),(105,109),(106,112),(108,110)], [(1,97),(2,102),(3,99),(4,104),(5,101),(6,98),(7,103),(8,100),(9,48),(10,45),(11,42),(12,47),(13,44),(14,41),(15,46),(16,43),(17,95),(18,92),(19,89),(20,94),(21,91),(22,96),(23,93),(24,90),(25,106),(26,111),(27,108),(28,105),(29,110),(30,107),(31,112),(32,109),(33,66),(34,71),(35,68),(36,65),(37,70),(38,67),(39,72),(40,69),(49,78),(50,75),(51,80),(52,77),(53,74),(54,79),(55,76),(56,73),(57,85),(58,82),(59,87),(60,84),(61,81),(62,86),(63,83),(64,88)]])
C7×C8.C22 is a maximal subgroup of
D28.39D4 M4(2).15D14 M4(2).16D14 D28.40D4 D56⋊C22 C56.C23 D28.44D4
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 28A | ··· | 28L | 28M | ··· | 28AD | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C7×D4 | C7×D4 | C8.C22 | C7×C8.C22 |
| kernel | C7×C8.C22 | C7×M4(2) | C7×SD16 | C7×Q16 | Q8×C14 | C7×C4○D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C28 | C2×C14 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 12 | 12 | 6 | 6 | 1 | 1 | 6 | 6 | 1 | 6 |
Matrix representation of C7×C8.C22 ►in GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 110 | 101 | 110 | 9 |
| 3 | 0 | 12 | 104 |
| 92 | 0 | 104 | 101 |
| 21 | 21 | 12 | 12 |
| 1 | 1 | 1 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 111 | 112 | 112 | 112 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[110,3,92,21,101,0,0,21,110,12,104,12,9,104,101,12],[1,0,0,0,1,112,0,0,1,0,112,0,0,0,0,1],[1,0,111,0,0,0,112,1,0,0,112,0,0,1,112,0] >;
C7×C8.C22 in GAP, Magma, Sage, TeX
C_7\times C_8.C_2^2
% in TeX
G:=Group("C7xC8.C2^2"); // GroupNames label
G:=SmallGroup(224,172);
// by ID
G=gap.SmallGroup(224,172);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,679,2090,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations