metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.D11, C22.11D4, C22:Dic11, C22.7D22, (C2xC22):2C4, C22.9(C2xC4), C11:2(C22:C4), (C2xDic11):2C2, C2.3(C11:D4), (C2xC22).7C22, (C22xC22).2C2, C2.5(C2xDic11), SmallGroup(176,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D11
G = < a,b,c,d,e | a2=b2=c2=d11=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, D11, Dic11, D22, C2xDic11, C11:D4, C23.D11
Smallest permutation representation of C23.D11
►On 88 points
Generators in S88
(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)
(1 32)(2 33)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 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 77 32 55)(2 76 33 54)(3 75 23 53)(4 74 24 52)(5 73 25 51)(6 72 26 50)(7 71 27 49)(8 70 28 48)(9 69 29 47)(10 68 30 46)(11 67 31 45)(12 86 34 64)(13 85 35 63)(14 84 36 62)(15 83 37 61)(16 82 38 60)(17 81 39 59)(18 80 40 58)(19 79 41 57)(20 78 42 56)(21 88 43 66)(22 87 44 65)
G:=sub<Sym(88)| (45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (1,32)(2,33)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (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), (1,77,32,55)(2,76,33,54)(3,75,23,53)(4,74,24,52)(5,73,25,51)(6,72,26,50)(7,71,27,49)(8,70,28,48)(9,69,29,47)(10,68,30,46)(11,67,31,45)(12,86,34,64)(13,85,35,63)(14,84,36,62)(15,83,37,61)(16,82,38,60)(17,81,39,59)(18,80,40,58)(19,79,41,57)(20,78,42,56)(21,88,43,66)(22,87,44,65)>;
G:=Group( (45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (1,32)(2,33)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (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), (1,77,32,55)(2,76,33,54)(3,75,23,53)(4,74,24,52)(5,73,25,51)(6,72,26,50)(7,71,27,49)(8,70,28,48)(9,69,29,47)(10,68,30,46)(11,67,31,45)(12,86,34,64)(13,85,35,63)(14,84,36,62)(15,83,37,61)(16,82,38,60)(17,81,39,59)(18,80,40,58)(19,79,41,57)(20,78,42,56)(21,88,43,66)(22,87,44,65) );
G=PermutationGroup([[(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88)], [(1,32),(2,33),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88)], [(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)], [(1,77,32,55),(2,76,33,54),(3,75,23,53),(4,74,24,52),(5,73,25,51),(6,72,26,50),(7,71,27,49),(8,70,28,48),(9,69,29,47),(10,68,30,46),(11,67,31,45),(12,86,34,64),(13,85,35,63),(14,84,36,62),(15,83,37,61),(16,82,38,60),(17,81,39,59),(18,80,40,58),(19,79,41,57),(20,78,42,56),(21,88,43,66),(22,87,44,65)]])
C23.D11 is a maximal subgroup of
C22.2D44 C23:Dic11 C23.11D22 C22:Dic22 C23.D22 C22:C4xD11 D22.D4 Dic11.D4 C44.48D4 C23.21D22 C4xC11:D4 C23.23D22 D4xDic11 C23.18D22 C44.17D4 C23:D22 C44:2D4 Dic11:D4 C24:D11
C23.D11 is a maximal quotient of
C44.55D4 C22.C42 D4:Dic11 C44.D4 C23:Dic11 Q8:Dic11 C44.10D4 C44.56D4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 11A | ··· | 11E | 22A | ··· | 22AI |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C4 | D4 | D11 | Dic11 | D22 | C11:D4 |
| kernel | C23.D11 | C2xDic11 | C22xC22 | C2xC22 | C22 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 5 | 10 | 5 | 20 |
Matrix representation of C23.D11 ►in GL3(F89) generated by
| 88 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 74 | 88 |
| 88 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 39 | 39 |
| 55 | 0 | 0 |
| 0 | 69 | 27 |
| 0 | 5 | 20 |
G:=sub<GL(3,GF(89))| [88,0,0,0,1,74,0,0,88],[88,0,0,0,1,0,0,0,1],[1,0,0,0,88,0,0,0,88],[1,0,0,0,16,39,0,0,39],[55,0,0,0,69,5,0,27,20] >;
C23.D11 in GAP, Magma, Sage, TeX
C_2^3.D_{11} % in TeX
G:=Group("C2^3.D11"); // GroupNames label
G:=SmallGroup(176,18);
// by ID
G=gap.SmallGroup(176,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,20,101,4004]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^11=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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