metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D44, C23.1D22, (C2×Dic11)⋊C4, (C22×D11)⋊C4, C11⋊1(C23⋊C4), C22⋊C4⋊1D11, (C2×C22).27D4, C2.4(D22⋊C4), C23.D11⋊1C2, C22.3(C4×D11), C22.2(C22⋊C4), C22.8(C11⋊D4), (C22×C22).5C22, (C2×C22).1(C2×C4), (C11×C22⋊C4)⋊1C2, (C2×C11⋊D4).1C2, SmallGroup(352,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.2D44
G = < a,b,c,d,e | a2=b2=c2=1, d22=a, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=bcd21 >
2C2
2C2
2C2
44C2
4C4
4C22
22C22
22C4
44C4
44C22
2C22
2C22
2C22
4D11
11C23
11C2×C4
22C2×C4
22D4
22D4
2D22
4D22
4C44
11C22⋊C4
11C2×D4
11C23⋊C4
Smallest permutation representation of C22.2D44
►On 88 points
Generators in S88
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(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 71)(2 24)(3 73)(4 26)(5 75)(6 28)(7 77)(8 30)(9 79)(10 32)(11 81)(12 34)(13 83)(14 36)(15 85)(16 38)(17 87)(18 40)(19 45)(20 42)(21 47)(22 44)(23 49)(25 51)(27 53)(29 55)(31 57)(33 59)(35 61)(37 63)(39 65)(41 67)(43 69)(46 68)(48 70)(50 72)(52 74)(54 76)(56 78)(58 80)(60 82)(62 84)(64 86)(66 88)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 45)(42 46)(43 47)(44 48)
(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 33)(2 80 50 32)(3 79)(4 30 52 78)(5 29)(6 76 54 28)(7 75)(8 26 56 74)(9 25)(10 72 58 24)(11 71)(12 22 60 70)(13 21)(14 68 62 20)(15 67)(16 18 64 66)(19 63)(23 59)(27 55)(31 51)(34 48 82 44)(35 47)(36 42 84 46)(37 41)(38 88 86 40)(39 87)(43 83)(45 85)(49 81)(53 77)(57 73)(61 69)
G:=sub<Sym(88)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(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), (1,71)(2,24)(3,73)(4,26)(5,75)(6,28)(7,77)(8,30)(9,79)(10,32)(11,81)(12,34)(13,83)(14,36)(15,85)(16,38)(17,87)(18,40)(19,45)(20,42)(21,47)(22,44)(23,49)(25,51)(27,53)(29,55)(31,57)(33,59)(35,61)(37,63)(39,65)(41,67)(43,69)(46,68)(48,70)(50,72)(52,74)(54,76)(56,78)(58,80)(60,82)(62,84)(64,86)(66,88), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,45)(42,46)(43,47)(44,48), (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), (1,33)(2,80,50,32)(3,79)(4,30,52,78)(5,29)(6,76,54,28)(7,75)(8,26,56,74)(9,25)(10,72,58,24)(11,71)(12,22,60,70)(13,21)(14,68,62,20)(15,67)(16,18,64,66)(19,63)(23,59)(27,55)(31,51)(34,48,82,44)(35,47)(36,42,84,46)(37,41)(38,88,86,40)(39,87)(43,83)(45,85)(49,81)(53,77)(57,73)(61,69)>;
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(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), (1,71)(2,24)(3,73)(4,26)(5,75)(6,28)(7,77)(8,30)(9,79)(10,32)(11,81)(12,34)(13,83)(14,36)(15,85)(16,38)(17,87)(18,40)(19,45)(20,42)(21,47)(22,44)(23,49)(25,51)(27,53)(29,55)(31,57)(33,59)(35,61)(37,63)(39,65)(41,67)(43,69)(46,68)(48,70)(50,72)(52,74)(54,76)(56,78)(58,80)(60,82)(62,84)(64,86)(66,88), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,45)(42,46)(43,47)(44,48), (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), (1,33)(2,80,50,32)(3,79)(4,30,52,78)(5,29)(6,76,54,28)(7,75)(8,26,56,74)(9,25)(10,72,58,24)(11,71)(12,22,60,70)(13,21)(14,68,62,20)(15,67)(16,18,64,66)(19,63)(23,59)(27,55)(31,51)(34,48,82,44)(35,47)(36,42,84,46)(37,41)(38,88,86,40)(39,87)(43,83)(45,85)(49,81)(53,77)(57,73)(61,69) );
G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(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)], [(1,71),(2,24),(3,73),(4,26),(5,75),(6,28),(7,77),(8,30),(9,79),(10,32),(11,81),(12,34),(13,83),(14,36),(15,85),(16,38),(17,87),(18,40),(19,45),(20,42),(21,47),(22,44),(23,49),(25,51),(27,53),(29,55),(31,57),(33,59),(35,61),(37,63),(39,65),(41,67),(43,69),(46,68),(48,70),(50,72),(52,74),(54,76),(56,78),(58,80),(60,82),(62,84),(64,86),(66,88)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,45),(42,46),(43,47),(44,48)], [(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)], [(1,33),(2,80,50,32),(3,79),(4,30,52,78),(5,29),(6,76,54,28),(7,75),(8,26,56,74),(9,25),(10,72,58,24),(11,71),(12,22,60,70),(13,21),(14,68,62,20),(15,67),(16,18,64,66),(19,63),(23,59),(27,55),(31,51),(34,48,82,44),(35,47),(36,42,84,46),(37,41),(38,88,86,40),(39,87),(43,83),(45,85),(49,81),(53,77),(57,73),(61,69)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 2 | 2 | 44 | 4 | 4 | 44 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D11 | D22 | C4×D11 | D44 | C11⋊D4 | C23⋊C4 | C22.2D44 |
| kernel | C22.2D44 | C23.D11 | C11×C22⋊C4 | C2×C11⋊D4 | C2×Dic11 | C22×D11 | C2×C22 | C22⋊C4 | C23 | C22 | C22 | C22 | C11 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 1 | 10 |
Matrix representation of C22.2D44 ►in GL4(𝔽89) generated by
| 61 | 58 | 0 | 0 |
| 31 | 28 | 0 | 0 |
| 0 | 0 | 61 | 58 |
| 0 | 0 | 31 | 28 |
| 28 | 31 | 0 | 0 |
| 58 | 61 | 0 | 0 |
| 73 | 53 | 61 | 58 |
| 36 | 72 | 31 | 28 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 35 | 59 | 76 | 76 |
| 30 | 49 | 13 | 88 |
| 25 | 64 | 54 | 30 |
| 25 | 7 | 59 | 40 |
| 51 | 51 | 0 | 0 |
| 45 | 38 | 0 | 0 |
| 9 | 34 | 25 | 64 |
| 12 | 80 | 82 | 64 |
G:=sub<GL(4,GF(89))| [61,31,0,0,58,28,0,0,0,0,61,31,0,0,58,28],[28,58,73,36,31,61,53,72,0,0,61,31,0,0,58,28],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[35,30,25,25,59,49,64,7,76,13,54,59,76,88,30,40],[51,45,9,12,51,38,34,80,0,0,25,82,0,0,64,64] >;
C22.2D44 in GAP, Magma, Sage, TeX
C_2^2._2D_{44} % in TeX
G:=Group("C2^2.2D44"); // GroupNames label
G:=SmallGroup(352,12);
// by ID
G=gap.SmallGroup(352,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,121,31,362,297,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^22=a,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^21>;
// generators/relations
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