direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.C8, C23.C56, C56.104D4, M5(2)⋊3C14, C28.34M4(2), (C2×C4).C56, (C2×C8).3C28, (C2×C28).2C8, (C2×C56).6C4, C8.24(C7×D4), (C22×C14).1C8, (C22×C28).9C4, C22.4(C2×C56), (C22×C4).4C28, C4.7(C7×M4(2)), (C7×M5(2))⋊11C2, C14.26(C22⋊C8), (C2×C56).309C22, C28.112(C22⋊C4), (C14×M4(2)).22C2, (C2×M4(2)).10C14, C2.7(C7×C22⋊C8), (C2×C4).67(C2×C28), (C2×C8).46(C2×C14), (C2×C14).22(C2×C8), C4.29(C7×C22⋊C4), (C2×C28).328(C2×C4), SmallGroup(448,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.C8
G = < a,b,c,d,e | a7=b2=c2=d2=1, e8=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 90 in 58 conjugacy classes, 34 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C16, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, M5(2), C2×M4(2), C56, C56, C2×C28, C2×C28, C22×C14, C23.C8, C112, C2×C56, C7×M4(2), C22×C28, C7×M5(2), C14×M4(2), C7×C23.C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C14, C22⋊C4, C2×C8, M4(2), C28, C2×C14, C22⋊C8, C56, C2×C28, C7×D4, C23.C8, C7×C22⋊C4, C2×C56, C7×M4(2), C7×C22⋊C8, C7×C23.C8
►On 112 points
Generators in S112
(1 66 82 102 18 35 61)(2 67 83 103 19 36 62)(3 68 84 104 20 37 63)(4 69 85 105 21 38 64)(5 70 86 106 22 39 49)(6 71 87 107 23 40 50)(7 72 88 108 24 41 51)(8 73 89 109 25 42 52)(9 74 90 110 26 43 53)(10 75 91 111 27 44 54)(11 76 92 112 28 45 55)(12 77 93 97 29 46 56)(13 78 94 98 30 47 57)(14 79 95 99 31 48 58)(15 80 96 100 32 33 59)(16 65 81 101 17 34 60)
(2 10)(3 11)(6 14)(7 15)(19 27)(20 28)(23 31)(24 32)(33 41)(36 44)(37 45)(40 48)(50 58)(51 59)(54 62)(55 63)(67 75)(68 76)(71 79)(72 80)(83 91)(84 92)(87 95)(88 96)(99 107)(100 108)(103 111)(104 112)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(97 105)(99 107)(101 109)(103 111)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)
(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)
G:=sub<Sym(112)| (1,66,82,102,18,35,61)(2,67,83,103,19,36,62)(3,68,84,104,20,37,63)(4,69,85,105,21,38,64)(5,70,86,106,22,39,49)(6,71,87,107,23,40,50)(7,72,88,108,24,41,51)(8,73,89,109,25,42,52)(9,74,90,110,26,43,53)(10,75,91,111,27,44,54)(11,76,92,112,28,45,55)(12,77,93,97,29,46,56)(13,78,94,98,30,47,57)(14,79,95,99,31,48,58)(15,80,96,100,32,33,59)(16,65,81,101,17,34,60), (2,10)(3,11)(6,14)(7,15)(19,27)(20,28)(23,31)(24,32)(33,41)(36,44)(37,45)(40,48)(50,58)(51,59)(54,62)(55,63)(67,75)(68,76)(71,79)(72,80)(83,91)(84,92)(87,95)(88,96)(99,107)(100,108)(103,111)(104,112), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112), (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)>;
G:=Group( (1,66,82,102,18,35,61)(2,67,83,103,19,36,62)(3,68,84,104,20,37,63)(4,69,85,105,21,38,64)(5,70,86,106,22,39,49)(6,71,87,107,23,40,50)(7,72,88,108,24,41,51)(8,73,89,109,25,42,52)(9,74,90,110,26,43,53)(10,75,91,111,27,44,54)(11,76,92,112,28,45,55)(12,77,93,97,29,46,56)(13,78,94,98,30,47,57)(14,79,95,99,31,48,58)(15,80,96,100,32,33,59)(16,65,81,101,17,34,60), (2,10)(3,11)(6,14)(7,15)(19,27)(20,28)(23,31)(24,32)(33,41)(36,44)(37,45)(40,48)(50,58)(51,59)(54,62)(55,63)(67,75)(68,76)(71,79)(72,80)(83,91)(84,92)(87,95)(88,96)(99,107)(100,108)(103,111)(104,112), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112), (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) );
G=PermutationGroup([[(1,66,82,102,18,35,61),(2,67,83,103,19,36,62),(3,68,84,104,20,37,63),(4,69,85,105,21,38,64),(5,70,86,106,22,39,49),(6,71,87,107,23,40,50),(7,72,88,108,24,41,51),(8,73,89,109,25,42,52),(9,74,90,110,26,43,53),(10,75,91,111,27,44,54),(11,76,92,112,28,45,55),(12,77,93,97,29,46,56),(13,78,94,98,30,47,57),(14,79,95,99,31,48,58),(15,80,96,100,32,33,59),(16,65,81,101,17,34,60)], [(2,10),(3,11),(6,14),(7,15),(19,27),(20,28),(23,31),(24,32),(33,41),(36,44),(37,45),(40,48),(50,58),(51,59),(54,62),(55,63),(67,75),(68,76),(71,79),(72,80),(83,91),(84,92),(87,95),(88,96),(99,107),(100,108),(103,111),(104,112)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48),(50,58),(52,60),(54,62),(56,64),(65,73),(67,75),(69,77),(71,79),(81,89),(83,91),(85,93),(87,95),(97,105),(99,107),(101,109),(103,111)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(97,105),(98,106),(99,107),(100,108),(101,109),(102,110),(103,111),(104,112)], [(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)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28X | 56A | ··· | 56X | 56Y | ··· | 56AJ | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C8 | C14 | C14 | C28 | C28 | C56 | C56 | D4 | M4(2) | C7×D4 | C7×M4(2) | C23.C8 | C7×C23.C8 |
| kernel | C7×C23.C8 | C7×M5(2) | C14×M4(2) | C2×C56 | C22×C28 | C23.C8 | C2×C28 | C22×C14 | M5(2) | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C23 | C56 | C28 | C8 | C4 | C7 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 4 | 4 | 12 | 6 | 12 | 12 | 24 | 24 | 2 | 2 | 12 | 12 | 2 | 12 |
Matrix representation of C7×C23.C8 ►in GL4(𝔽113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 30 | 0 | 98 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 30 | 100 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 |
| 30 | 100 | 98 | 111 |
| 0 | 1 | 0 | 0 |
| 95 | 77 | 56 | 13 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[1,0,0,30,0,112,0,0,0,0,1,98,0,0,0,112],[1,0,0,30,0,1,0,100,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[0,30,0,95,0,100,1,77,1,98,0,56,0,111,0,13] >;
C7×C23.C8 in GAP, Magma, Sage, TeX
C_7\times C_2^3.C_8
% in TeX
G:=Group("C7xC2^3.C8"); // GroupNames label
G:=SmallGroup(448,153);
// by ID
G=gap.SmallGroup(448,153);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,7059,4911,102,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^2=1,e^8=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations