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## G = C2×Dic11order 88 = 23·11

### Direct product of C2 and Dic11

Aliases: C2×Dic11, C22⋊C4, C2.2D22, C22.D11, C22.4C22, C112(C2×C4), (C2×C22).C2, SmallGroup(88,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C2×Dic11
 Chief series C1 — C11 — C22 — Dic11 — C2×Dic11
 Lower central C11 — C2×Dic11
 Upper central C1 — C22

Generators and relations for C2×Dic11
G = < a,b,c | a2=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×Dic11

 class 1 2A 2B 2C 4A 4B 4C 4D 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E 22F 22G 22H 22I 22J 22K 22L 22M 22N 22O size 1 1 1 1 11 11 11 11 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 -1 1 linear of order 2 ρ3 1 1 -1 -1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 -1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 -1 1 -1 i i -i -i 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 linear of order 4 ρ6 1 -1 -1 1 i -i -i i 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 -1 -1 1 -i i i -i 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ8 1 -1 1 -1 -i -i i i 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 linear of order 4 ρ9 2 2 2 2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 orthogonal lifted from D11 ρ10 2 2 -2 -2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 ζ1110+ζ11 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 ζ117+ζ114 ζ119+ζ112 -ζ118-ζ113 ζ118+ζ113 orthogonal lifted from D22 ρ11 2 2 -2 -2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 ζ119+ζ112 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 ζ118+ζ113 ζ117+ζ114 -ζ116-ζ115 ζ116+ζ115 orthogonal lifted from D22 ρ12 2 2 -2 -2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 ζ116+ζ115 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 ζ119+ζ112 ζ1110+ζ11 -ζ117-ζ114 ζ117+ζ114 orthogonal lifted from D22 ρ13 2 2 2 2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 orthogonal lifted from D11 ρ14 2 2 2 2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 orthogonal lifted from D11 ρ15 2 2 2 2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 orthogonal lifted from D11 ρ16 2 2 -2 -2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 ζ117+ζ114 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 ζ116+ζ115 ζ118+ζ113 -ζ1110-ζ11 ζ1110+ζ11 orthogonal lifted from D22 ρ17 2 2 -2 -2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 ζ118+ζ113 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 ζ1110+ζ11 ζ116+ζ115 -ζ119-ζ112 ζ119+ζ112 orthogonal lifted from D22 ρ18 2 2 2 2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 orthogonal lifted from D11 ρ19 2 -2 2 -2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ118-ζ113 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ116-ζ115 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ119-ζ112 -ζ1110-ζ11 ζ117+ζ114 -ζ117-ζ114 symplectic lifted from Dic11, Schur index 2 ρ20 2 -2 2 -2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ1110-ζ11 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ119-ζ112 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ118-ζ113 -ζ117-ζ114 ζ116+ζ115 -ζ116-ζ115 symplectic lifted from Dic11, Schur index 2 ρ21 2 -2 2 -2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ117-ζ114 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ118-ζ113 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ1110-ζ11 -ζ116-ζ115 ζ119+ζ112 -ζ119-ζ112 symplectic lifted from Dic11, Schur index 2 ρ22 2 -2 -2 2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ117-ζ114 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ118-ζ113 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ119-ζ112 symplectic lifted from Dic11, Schur index 2 ρ23 2 -2 -2 2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ119-ζ112 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ117-ζ114 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ1110-ζ11 symplectic lifted from Dic11, Schur index 2 ρ24 2 -2 -2 2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ116-ζ115 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ1110-ζ11 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ118-ζ113 symplectic lifted from Dic11, Schur index 2 ρ25 2 -2 -2 2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ1110-ζ11 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ119-ζ112 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ116-ζ115 symplectic lifted from Dic11, Schur index 2 ρ26 2 -2 2 -2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ119-ζ112 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ117-ζ114 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ116-ζ115 -ζ118-ζ113 ζ1110+ζ11 -ζ1110-ζ11 symplectic lifted from Dic11, Schur index 2 ρ27 2 -2 -2 2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ118-ζ113 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ116-ζ115 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ117-ζ114 symplectic lifted from Dic11, Schur index 2 ρ28 2 -2 2 -2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ116-ζ115 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ1110-ζ11 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ117-ζ114 -ζ119-ζ112 ζ118+ζ113 -ζ118-ζ113 symplectic lifted from Dic11, Schur index 2

Smallest permutation representation of C2×Dic11
Regular action on 88 points
Generators in S88
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(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 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 45 12 56)(2 66 13 55)(3 65 14 54)(4 64 15 53)(5 63 16 52)(6 62 17 51)(7 61 18 50)(8 60 19 49)(9 59 20 48)(10 58 21 47)(11 57 22 46)(23 75 34 86)(24 74 35 85)(25 73 36 84)(26 72 37 83)(27 71 38 82)(28 70 39 81)(29 69 40 80)(30 68 41 79)(31 67 42 78)(32 88 43 77)(33 87 44 76)

G:=sub<Sym(88)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(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), (1,45,12,56)(2,66,13,55)(3,65,14,54)(4,64,15,53)(5,63,16,52)(6,62,17,51)(7,61,18,50)(8,60,19,49)(9,59,20,48)(10,58,21,47)(11,57,22,46)(23,75,34,86)(24,74,35,85)(25,73,36,84)(26,72,37,83)(27,71,38,82)(28,70,39,81)(29,69,40,80)(30,68,41,79)(31,67,42,78)(32,88,43,77)(33,87,44,76)>;

G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(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), (1,45,12,56)(2,66,13,55)(3,65,14,54)(4,64,15,53)(5,63,16,52)(6,62,17,51)(7,61,18,50)(8,60,19,49)(9,59,20,48)(10,58,21,47)(11,57,22,46)(23,75,34,86)(24,74,35,85)(25,73,36,84)(26,72,37,83)(27,71,38,82)(28,70,39,81)(29,69,40,80)(30,68,41,79)(31,67,42,78)(32,88,43,77)(33,87,44,76) );

G=PermutationGroup([(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22),(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)], [(1,45,12,56),(2,66,13,55),(3,65,14,54),(4,64,15,53),(5,63,16,52),(6,62,17,51),(7,61,18,50),(8,60,19,49),(9,59,20,48),(10,58,21,47),(11,57,22,46),(23,75,34,86),(24,74,35,85),(25,73,36,84),(26,72,37,83),(27,71,38,82),(28,70,39,81),(29,69,40,80),(30,68,41,79),(31,67,42,78),(32,88,43,77),(33,87,44,76)])

C2×Dic11 is a maximal subgroup of   Dic11⋊C4  C44⋊C4  D22⋊C4  C23.D11  C2×C4×D11  D42D11
C2×Dic11 is a maximal quotient of   C44.C4  C44⋊C4  C23.D11

Matrix representation of C2×Dic11 in GL4(𝔽89) generated by

 1 0 0 0 0 88 0 0 0 0 1 0 0 0 0 1
,
 88 0 0 0 0 1 0 0 0 0 88 1 0 0 80 8
,
 34 0 0 0 0 88 0 0 0 0 9 74 0 0 35 80
G:=sub<GL(4,GF(89))| [1,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[88,0,0,0,0,1,0,0,0,0,88,80,0,0,1,8],[34,0,0,0,0,88,0,0,0,0,9,35,0,0,74,80] >;

C2×Dic11 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{11}
% in TeX

G:=Group("C2xDic11");
// GroupNames label

G:=SmallGroup(88,6);
// by ID

G=gap.SmallGroup(88,6);
# by ID

G:=PCGroup([4,-2,-2,-2,-11,16,1283]);
// Polycyclic

G:=Group<a,b,c|a^2=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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