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G = C2×Dic11order 88 = 23·11

Direct product of C2 and Dic11

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C11 — C2×Dic11
C1C11C22Dic11 — C2×Dic11
C11 — C2×Dic11
C1C22

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

11C4
11C4
11C2×C4

Character table of C2×Dic11

 class 12A2B2C4A4B4C4D11A11B11C11D11E22A22B22C22D22E22F22G22H22I22J22K22L22M22N22O
 size 11111111111122222222222222222222
ρ11111111111111111111111111111    trivial
ρ211-1-11-11-1111111-1-1-1-11-1-1-1-1-111-11    linear of order 2
ρ311-1-1-11-11111111-1-1-1-11-1-1-1-1-111-11    linear of order 2
ρ41111-1-1-1-111111111111111111111    linear of order 2
ρ51-11-1ii-i-i11111-11111-1-1-1-1-1-1-1-11-1    linear of order 4
ρ61-1-11i-i-ii11111-1-1-1-1-1-111111-1-1-1-1    linear of order 4
ρ71-1-11-iii-i11111-1-1-1-1-1-111111-1-1-1-1    linear of order 4
ρ81-11-1-i-iii11111-11111-1-1-1-1-1-1-1-11-1    linear of order 4
ρ922220000ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ117114ζ116115ζ118113ζ111011ζ118113ζ119112ζ117114ζ116115ζ118113ζ111011ζ111011ζ116115ζ119112ζ119112    orthogonal lifted from D11
ρ1022-2-20000ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115116115119112111011117114ζ111011118113116115119112111011117114ζ117114ζ119112118113ζ118113    orthogonal lifted from D22
ρ1122-2-20000ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011111011117114119112118113ζ119112116115111011117114119112118113ζ118113ζ117114116115ζ116115    orthogonal lifted from D22
ρ1222-2-20000ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113118113111011116115119112ζ116115117114118113111011116115119112ζ119112ζ111011117114ζ117114    orthogonal lifted from D22
ρ1322220000ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ111011ζ117114ζ119112ζ118113ζ119112ζ116115ζ111011ζ117114ζ119112ζ118113ζ118113ζ117114ζ116115ζ116115    orthogonal lifted from D11
ρ1422220000ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ118113ζ111011ζ116115ζ119112ζ116115ζ117114ζ118113ζ111011ζ116115ζ119112ζ119112ζ111011ζ117114ζ117114    orthogonal lifted from D11
ρ1522220000ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ119112ζ118113ζ117114ζ116115ζ117114ζ111011ζ119112ζ118113ζ117114ζ116115ζ116115ζ118113ζ111011ζ111011    orthogonal lifted from D11
ρ1622-2-20000ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112119112118113117114116115ζ117114111011119112118113117114116115ζ116115ζ118113111011ζ111011    orthogonal lifted from D22
ρ1722-2-20000ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114117114116115118113111011ζ118113119112117114116115118113111011ζ111011ζ116115119112ζ119112    orthogonal lifted from D22
ρ1822220000ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ116115ζ119112ζ111011ζ117114ζ111011ζ118113ζ116115ζ119112ζ111011ζ117114ζ117114ζ119112ζ118113ζ118113    orthogonal lifted from D11
ρ192-22-20000ζ117114ζ118113ζ111011ζ116115ζ119112118113ζ118113ζ111011ζ116115ζ119112116115117114118113111011116115119112119112111011ζ117114117114    symplectic lifted from Dic11, Schur index 2
ρ202-22-20000ζ116115ζ111011ζ117114ζ119112ζ118113111011ζ111011ζ117114ζ119112ζ118113119112116115111011117114119112118113118113117114ζ116115116115    symplectic lifted from Dic11, Schur index 2
ρ212-22-20000ζ119112ζ117114ζ116115ζ118113ζ111011117114ζ117114ζ116115ζ118113ζ111011118113119112117114116115118113111011111011116115ζ119112119112    symplectic lifted from Dic11, Schur index 2
ρ222-2-220000ζ119112ζ117114ζ116115ζ118113ζ111011117114117114116115118113111011118113ζ119112ζ117114ζ116115ζ118113ζ111011111011116115119112119112    symplectic lifted from Dic11, Schur index 2
ρ232-2-220000ζ111011ζ119112ζ118113ζ117114ζ116115119112119112118113117114116115117114ζ111011ζ119112ζ118113ζ117114ζ116115116115118113111011111011    symplectic lifted from Dic11, Schur index 2
ρ242-2-220000ζ118113ζ116115ζ119112ζ111011ζ117114116115116115119112111011117114111011ζ118113ζ116115ζ119112ζ111011ζ117114117114119112118113118113    symplectic lifted from Dic11, Schur index 2
ρ252-2-220000ζ116115ζ111011ζ117114ζ119112ζ118113111011111011117114119112118113119112ζ116115ζ111011ζ117114ζ119112ζ118113118113117114116115116115    symplectic lifted from Dic11, Schur index 2
ρ262-22-20000ζ111011ζ119112ζ118113ζ117114ζ116115119112ζ119112ζ118113ζ117114ζ116115117114111011119112118113117114116115116115118113ζ111011111011    symplectic lifted from Dic11, Schur index 2
ρ272-2-220000ζ117114ζ118113ζ111011ζ116115ζ119112118113118113111011116115119112116115ζ117114ζ118113ζ111011ζ116115ζ119112119112111011117114117114    symplectic lifted from Dic11, Schur index 2
ρ282-22-20000ζ118113ζ116115ζ119112ζ111011ζ117114116115ζ116115ζ119112ζ111011ζ117114111011118113116115119112111011117114117114119112ζ118113118113    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

1000
08800
0010
0001
,
88000
0100
00881
00808
,
34000
08800
00974
003580
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

Export

Subgroup lattice of C2×Dic11 in TeX
Character table of C2×Dic11 in TeX

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