direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C4○D4, D4⋊2C22, Q8⋊2C22, C22.13C23, C44.21C22, (C2×C44)⋊7C2, (C2×C4)⋊3C22, C44○(D4×C11), C44○(Q8×C11), (D4×C11)⋊5C2, C4.5(C2×C22), (Q8×C11)⋊5C2, C22.(C2×C22), (C2×C22).2C22, C2.3(C22×C22), SmallGroup(176,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C4○D4
G = < a,b,c,d | a11=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Smallest permutation representation of C11×C4○D4
►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 49 27 38)(2 50 28 39)(3 51 29 40)(4 52 30 41)(5 53 31 42)(6 54 32 43)(7 55 33 44)(8 45 23 34)(9 46 24 35)(10 47 25 36)(11 48 26 37)(12 74 85 63)(13 75 86 64)(14 76 87 65)(15 77 88 66)(16 67 78 56)(17 68 79 57)(18 69 80 58)(19 70 81 59)(20 71 82 60)(21 72 83 61)(22 73 84 62)
(1 38 27 49)(2 39 28 50)(3 40 29 51)(4 41 30 52)(5 42 31 53)(6 43 32 54)(7 44 33 55)(8 34 23 45)(9 35 24 46)(10 36 25 47)(11 37 26 48)(12 74 85 63)(13 75 86 64)(14 76 87 65)(15 77 88 66)(16 67 78 56)(17 68 79 57)(18 69 80 58)(19 70 81 59)(20 71 82 60)(21 72 83 61)(22 73 84 62)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 78)(9 79)(10 80)(11 81)(12 30)(13 31)(14 32)(15 33)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(22 29)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)
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), (1,49,27,38)(2,50,28,39)(3,51,29,40)(4,52,30,41)(5,53,31,42)(6,54,32,43)(7,55,33,44)(8,45,23,34)(9,46,24,35)(10,47,25,36)(11,48,26,37)(12,74,85,63)(13,75,86,64)(14,76,87,65)(15,77,88,66)(16,67,78,56)(17,68,79,57)(18,69,80,58)(19,70,81,59)(20,71,82,60)(21,72,83,61)(22,73,84,62), (1,38,27,49)(2,39,28,50)(3,40,29,51)(4,41,30,52)(5,42,31,53)(6,43,32,54)(7,44,33,55)(8,34,23,45)(9,35,24,46)(10,36,25,47)(11,37,26,48)(12,74,85,63)(13,75,86,64)(14,76,87,65)(15,77,88,66)(16,67,78,56)(17,68,79,57)(18,69,80,58)(19,70,81,59)(20,71,82,60)(21,72,83,61)(22,73,84,62), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,78)(9,79)(10,80)(11,81)(12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)>;
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), (1,49,27,38)(2,50,28,39)(3,51,29,40)(4,52,30,41)(5,53,31,42)(6,54,32,43)(7,55,33,44)(8,45,23,34)(9,46,24,35)(10,47,25,36)(11,48,26,37)(12,74,85,63)(13,75,86,64)(14,76,87,65)(15,77,88,66)(16,67,78,56)(17,68,79,57)(18,69,80,58)(19,70,81,59)(20,71,82,60)(21,72,83,61)(22,73,84,62), (1,38,27,49)(2,39,28,50)(3,40,29,51)(4,41,30,52)(5,42,31,53)(6,43,32,54)(7,44,33,55)(8,34,23,45)(9,35,24,46)(10,36,25,47)(11,37,26,48)(12,74,85,63)(13,75,86,64)(14,76,87,65)(15,77,88,66)(16,67,78,56)(17,68,79,57)(18,69,80,58)(19,70,81,59)(20,71,82,60)(21,72,83,61)(22,73,84,62), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,78)(9,79)(10,80)(11,81)(12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66) );
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)], [(1,49,27,38),(2,50,28,39),(3,51,29,40),(4,52,30,41),(5,53,31,42),(6,54,32,43),(7,55,33,44),(8,45,23,34),(9,46,24,35),(10,47,25,36),(11,48,26,37),(12,74,85,63),(13,75,86,64),(14,76,87,65),(15,77,88,66),(16,67,78,56),(17,68,79,57),(18,69,80,58),(19,70,81,59),(20,71,82,60),(21,72,83,61),(22,73,84,62)], [(1,38,27,49),(2,39,28,50),(3,40,29,51),(4,41,30,52),(5,42,31,53),(6,43,32,54),(7,44,33,55),(8,34,23,45),(9,35,24,46),(10,36,25,47),(11,37,26,48),(12,74,85,63),(13,75,86,64),(14,76,87,65),(15,77,88,66),(16,67,78,56),(17,68,79,57),(18,69,80,58),(19,70,81,59),(20,71,82,60),(21,72,83,61),(22,73,84,62)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,78),(9,79),(10,80),(11,81),(12,30),(13,31),(14,32),(15,33),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(22,29),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66)]])
C11×C4○D4 is a maximal subgroup of
C44.56D4 Q8.Dic11 Q8⋊D22 D4.8D22 D4.9D22 D4⋊8D22 D4.10D22
C11×C4○D4 is a maximal quotient of D4×C44 Q8×C44
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AN | 44A | ··· | 44T | 44U | ··· | 44AX |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | C4○D4 | C11×C4○D4 |
| kernel | C11×C4○D4 | C2×C44 | D4×C11 | Q8×C11 | C4○D4 | C2×C4 | D4 | Q8 | C11 | C1 |
| # reps | 1 | 3 | 3 | 1 | 10 | 30 | 30 | 10 | 2 | 20 |
Matrix representation of C11×C4○D4 ►in GL2(𝔽89) generated by
| 64 | 0 |
| 0 | 64 |
| 55 | 0 |
| 0 | 55 |
| 34 | 0 |
| 0 | 55 |
| 0 | 55 |
| 34 | 0 |
G:=sub<GL(2,GF(89))| [64,0,0,64],[55,0,0,55],[34,0,0,55],[0,34,55,0] >;
C11×C4○D4 in GAP, Magma, Sage, TeX
C_{11}\times C_4\circ D_4 % in TeX
G:=Group("C11xC4oD4"); // GroupNames label
G:=SmallGroup(176,40);
// by ID
G=gap.SmallGroup(176,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-11,-2,901,342]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations
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