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## G = C23⋊Dic11order 352 = 25·11

### The semidirect product of C23 and Dic11 acting via Dic11/C11=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C23⋊Dic11
 Chief series C1 — C11 — C22 — C2×C22 — C22×C22 — C23.D11 — C23⋊Dic11
 Lower central C11 — C22 — C2×C22 — C23⋊Dic11
 Upper central C1 — C2 — C23 — C2×D4

Generators and relations for C23⋊Dic11
G = < a,b,c,d,e | a2=b2=c2=d22=1, e2=d11, ab=ba, dad-1=ac=ca, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Smallest permutation representation of C23⋊Dic11
On 88 points
Generators in S88
(1 85)(2 31)(3 87)(4 33)(5 67)(6 35)(7 69)(8 37)(9 71)(10 39)(11 73)(12 41)(13 75)(14 43)(15 77)(16 23)(17 79)(18 25)(19 81)(20 27)(21 83)(22 29)(24 53)(26 55)(28 57)(30 59)(32 61)(34 63)(36 65)(38 45)(40 47)(42 49)(44 51)(46 72)(48 74)(50 76)(52 78)(54 80)(56 82)(58 84)(60 86)(62 88)(64 68)(66 70)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)
(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 85 12 74)(2 84 13 73)(3 83 14 72)(4 82 15 71)(5 81 16 70)(6 80 17 69)(7 79 18 68)(8 78 19 67)(9 77 20 88)(10 76 21 87)(11 75 22 86)(23 55 34 66)(24 54 35 65)(25 53 36 64)(26 52 37 63)(27 51 38 62)(28 50 39 61)(29 49 40 60)(30 48 41 59)(31 47 42 58)(32 46 43 57)(33 45 44 56)

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J 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 4 4 44 44 44 44 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

61 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - - + + image C1 C2 C2 C4 C4 D4 D11 Dic11 Dic11 D22 C11⋊D4 C23⋊C4 C23⋊Dic11 kernel C23⋊Dic11 C23.D11 D4×C22 C2×C44 C22×C22 C2×C22 C2×D4 C2×C4 C23 C23 C22 C11 C1 # reps 1 2 1 2 2 2 5 5 5 5 20 1 10

Matrix representation of C23⋊Dic11 in GL4(𝔽89) generated by

 61 31 0 41 2 0 48 50 58 29 53 54 38 19 31 64
,
 12 50 0 0 63 77 0 0 38 19 3 50 59 74 39 86
,
 88 0 0 0 0 88 0 0 0 0 88 0 0 0 0 88
,
 26 86 0 0 87 31 0 0 49 82 50 3 35 33 86 71
,
 49 68 26 34 73 51 44 64 61 12 12 4 80 75 80 66
G:=sub<GL(4,GF(89))| [61,2,58,38,31,0,29,19,0,48,53,31,41,50,54,64],[12,63,38,59,50,77,19,74,0,0,3,39,0,0,50,86],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[26,87,49,35,86,31,82,33,0,0,50,86,0,0,3,71],[49,73,61,80,68,51,12,75,26,44,12,80,34,64,4,66] >;

C23⋊Dic11 in GAP, Magma, Sage, TeX

C_2^3\rtimes {\rm Dic}_{11}
% in TeX

G:=Group("C2^3:Dic11");
// GroupNames label

G:=SmallGroup(352,40);
// by ID

G=gap.SmallGroup(352,40);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,188,579,11525]);
// Polycyclic

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

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