direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×M4(2), C4.C44, C8⋊3C22, C88⋊7C2, C44.4C4, C22.C44, C44.22C22, (C2×C22).1C4, (C2×C44).8C2, C2.3(C2×C44), (C2×C4).2C22, C4.6(C2×C22), C22.12(C2×C4), SmallGroup(176,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×M4(2)
G = < a,b,c | a11=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of C11×M4(2)
►On 88 points
Generators in S88
(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)
(1 79 67 30 53 40 59 14)(2 80 68 31 54 41 60 15)(3 81 69 32 55 42 61 16)(4 82 70 33 45 43 62 17)(5 83 71 23 46 44 63 18)(6 84 72 24 47 34 64 19)(7 85 73 25 48 35 65 20)(8 86 74 26 49 36 66 21)(9 87 75 27 50 37 56 22)(10 88 76 28 51 38 57 12)(11 78 77 29 52 39 58 13)
(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 23)(19 24)(20 25)(21 26)(22 27)(34 84)(35 85)(36 86)(37 87)(38 88)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)
G:=sub<Sym(88)| (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), (1,79,67,30,53,40,59,14)(2,80,68,31,54,41,60,15)(3,81,69,32,55,42,61,16)(4,82,70,33,45,43,62,17)(5,83,71,23,46,44,63,18)(6,84,72,24,47,34,64,19)(7,85,73,25,48,35,65,20)(8,86,74,26,49,36,66,21)(9,87,75,27,50,37,56,22)(10,88,76,28,51,38,57,12)(11,78,77,29,52,39,58,13), (12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,23)(19,24)(20,25)(21,26)(22,27)(34,84)(35,85)(36,86)(37,87)(38,88)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)>;
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), (1,79,67,30,53,40,59,14)(2,80,68,31,54,41,60,15)(3,81,69,32,55,42,61,16)(4,82,70,33,45,43,62,17)(5,83,71,23,46,44,63,18)(6,84,72,24,47,34,64,19)(7,85,73,25,48,35,65,20)(8,86,74,26,49,36,66,21)(9,87,75,27,50,37,56,22)(10,88,76,28,51,38,57,12)(11,78,77,29,52,39,58,13), (12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,23)(19,24)(20,25)(21,26)(22,27)(34,84)(35,85)(36,86)(37,87)(38,88)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83) );
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)], [(1,79,67,30,53,40,59,14),(2,80,68,31,54,41,60,15),(3,81,69,32,55,42,61,16),(4,82,70,33,45,43,62,17),(5,83,71,23,46,44,63,18),(6,84,72,24,47,34,64,19),(7,85,73,25,48,35,65,20),(8,86,74,26,49,36,66,21),(9,87,75,27,50,37,56,22),(10,88,76,28,51,38,57,12),(11,78,77,29,52,39,58,13)], [(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,23),(19,24),(20,25),(21,26),(22,27),(34,84),(35,85),(36,86),(37,87),(38,88),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83)]])
C11×M4(2) is a maximal subgroup of
C44.53D4 C44.46D4 C44.47D4 D44⋊4C4 D44.C4 C8⋊D22 C8.D22
110 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22T | 44A | ··· | 44T | 44U | ··· | 44AD | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C11 | C22 | C22 | C44 | C44 | M4(2) | C11×M4(2) |
| kernel | C11×M4(2) | C88 | C2×C44 | C44 | C2×C22 | M4(2) | C8 | C2×C4 | C4 | C22 | C11 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 10 | 20 | 10 | 20 | 20 | 2 | 20 |
Matrix representation of C11×M4(2) ►in GL3(𝔽89) generated by
| 45 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 34 | 0 |
| 88 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 88 |
G:=sub<GL(3,GF(89))| [45,0,0,0,1,0,0,0,1],[1,0,0,0,0,34,0,1,0],[88,0,0,0,1,0,0,0,88] >;
C11×M4(2) in GAP, Magma, Sage, TeX
C_{11}\times M_4(2) % in TeX
G:=Group("C11xM4(2)"); // GroupNames label
G:=SmallGroup(176,23);
// by ID
G=gap.SmallGroup(176,23);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-2,220,901,58]);
// Polycyclic
G:=Group<a,b,c|a^11=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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