direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7xC22:C8, C22:C56, C28.65D4, C23.2C28, C14.8M4(2), (C2xC14):1C8, (C2xC56):3C2, (C2xC8):1C14, (C2xC4).3C28, C2.1(C2xC56), (C2xC28).6C4, C4.16(C7xD4), C14.11(C2xC8), (C22xC4).2C14, C22.9(C2xC28), (C22xC14).3C4, (C22xC28).3C2, C2.2(C7xM4(2)), C14.20(C22:C4), (C2xC28).135C22, C2.2(C7xC22:C4), (C2xC4).31(C2xC14), (C2xC14).38(C2xC4), SmallGroup(224,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7xC22:C8
G = < a,b,c,d | a7=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Quotients: C1, C2, C4, C22, C7, C8, C2xC4, D4, C14, C22:C4, C2xC8, M4(2), C28, C2xC14, C22:C8, C56, C2xC28, C7xD4, C7xC22:C4, C2xC56, C7xM4(2), C7xC22:C8
Smallest permutation representation of C7xC22:C8
►On 112 points
Generators in S112
(1 69 16 81 25 73 17)(2 70 9 82 26 74 18)(3 71 10 83 27 75 19)(4 72 11 84 28 76 20)(5 65 12 85 29 77 21)(6 66 13 86 30 78 22)(7 67 14 87 31 79 23)(8 68 15 88 32 80 24)(33 61 105 49 97 41 89)(34 62 106 50 98 42 90)(35 63 107 51 99 43 91)(36 64 108 52 100 44 92)(37 57 109 53 101 45 93)(38 58 110 54 102 46 94)(39 59 111 55 103 47 95)(40 60 112 56 104 48 96)
(2 40)(4 34)(6 36)(8 38)(9 112)(11 106)(13 108)(15 110)(18 96)(20 90)(22 92)(24 94)(26 104)(28 98)(30 100)(32 102)(42 76)(44 78)(46 80)(48 74)(50 84)(52 86)(54 88)(56 82)(58 68)(60 70)(62 72)(64 66)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 112)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 95)(18 96)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 65)(64 66)
(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)
G:=sub<Sym(112)| (1,69,16,81,25,73,17)(2,70,9,82,26,74,18)(3,71,10,83,27,75,19)(4,72,11,84,28,76,20)(5,65,12,85,29,77,21)(6,66,13,86,30,78,22)(7,67,14,87,31,79,23)(8,68,15,88,32,80,24)(33,61,105,49,97,41,89)(34,62,106,50,98,42,90)(35,63,107,51,99,43,91)(36,64,108,52,100,44,92)(37,57,109,53,101,45,93)(38,58,110,54,102,46,94)(39,59,111,55,103,47,95)(40,60,112,56,104,48,96), (2,40)(4,34)(6,36)(8,38)(9,112)(11,106)(13,108)(15,110)(18,96)(20,90)(22,92)(24,94)(26,104)(28,98)(30,100)(32,102)(42,76)(44,78)(46,80)(48,74)(50,84)(52,86)(54,88)(56,82)(58,68)(60,70)(62,72)(64,66), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,112)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,95)(18,96)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,65)(64,66), (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)>;
G:=Group( (1,69,16,81,25,73,17)(2,70,9,82,26,74,18)(3,71,10,83,27,75,19)(4,72,11,84,28,76,20)(5,65,12,85,29,77,21)(6,66,13,86,30,78,22)(7,67,14,87,31,79,23)(8,68,15,88,32,80,24)(33,61,105,49,97,41,89)(34,62,106,50,98,42,90)(35,63,107,51,99,43,91)(36,64,108,52,100,44,92)(37,57,109,53,101,45,93)(38,58,110,54,102,46,94)(39,59,111,55,103,47,95)(40,60,112,56,104,48,96), (2,40)(4,34)(6,36)(8,38)(9,112)(11,106)(13,108)(15,110)(18,96)(20,90)(22,92)(24,94)(26,104)(28,98)(30,100)(32,102)(42,76)(44,78)(46,80)(48,74)(50,84)(52,86)(54,88)(56,82)(58,68)(60,70)(62,72)(64,66), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,112)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,95)(18,96)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,103)(26,104)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,65)(64,66), (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) );
G=PermutationGroup([[(1,69,16,81,25,73,17),(2,70,9,82,26,74,18),(3,71,10,83,27,75,19),(4,72,11,84,28,76,20),(5,65,12,85,29,77,21),(6,66,13,86,30,78,22),(7,67,14,87,31,79,23),(8,68,15,88,32,80,24),(33,61,105,49,97,41,89),(34,62,106,50,98,42,90),(35,63,107,51,99,43,91),(36,64,108,52,100,44,92),(37,57,109,53,101,45,93),(38,58,110,54,102,46,94),(39,59,111,55,103,47,95),(40,60,112,56,104,48,96)], [(2,40),(4,34),(6,36),(8,38),(9,112),(11,106),(13,108),(15,110),(18,96),(20,90),(22,92),(24,94),(26,104),(28,98),(30,100),(32,102),(42,76),(44,78),(46,80),(48,74),(50,84),(52,86),(54,88),(56,82),(58,68),(60,70),(62,72),(64,66)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,112),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,95),(18,96),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(25,103),(26,104),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,73),(48,74),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,81),(56,82),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,65),(64,66)], [(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)]])
C7xC22:C8 is a maximal subgroup of
C23.30D28 (C22xD7):C8 (C2xDic7):C8 C22.2D56 Dic7.5M4(2) Dic7.M4(2) C56:C4:C2 C23.34D28 C23.35D28 C23.10D28 C7:D4:C8 D14:M4(2) D14:C8:C2 D14:2M4(2) Dic7:M4(2) C7:C8:26D4 D28.31D4 D28:13D4 D28.32D4 D28:14D4 C23.38D28 C22.D56 C23.13D28 Dic14:14D4 C22:Dic28 D4xC56
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C28 | C28 | C56 | D4 | M4(2) | C7xD4 | C7xM4(2) |
| kernel | C7xC22:C8 | C2xC56 | C22xC28 | C2xC28 | C22xC14 | C22:C8 | C2xC14 | C2xC8 | C22xC4 | C2xC4 | C23 | C22 | C28 | C14 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 8 | 12 | 6 | 12 | 12 | 48 | 2 | 2 | 12 | 12 |
Matrix representation of C7xC22:C8 ►in GL4(F113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 52 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 52 | 111 |
| 0 | 0 | 53 | 61 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,52,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,44,0,0,0,0,52,53,0,0,111,61] >;
C7xC22:C8 in GAP, Magma, Sage, TeX
C_7\times C_2^2\rtimes C_8
% in TeX
G:=Group("C7xC2^2:C8"); // GroupNames label
G:=SmallGroup(224,47);
// by ID
G=gap.SmallGroup(224,47);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,88]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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