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## G = C11×C23⋊C4order 352 = 25·11

### Direct product of C11 and C23⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C11×C23⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C22 — C11×C22⋊C4 — C11×C23⋊C4
 Lower central C1 — C2 — C22 — C11×C23⋊C4
 Upper central C1 — C22 — C22×C22 — C11×C23⋊C4

Generators and relations for C11×C23⋊C4
G = < a,b,c,d,e | a11=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Smallest permutation representation of C11×C23⋊C4
On 88 points
Generators in S88
(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 30)(2 31)(3 32)(4 33)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 54)(13 55)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 67)(44 68)(56 85)(57 86)(58 87)(59 88)(60 78)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 56)(8 57)(9 58)(10 59)(11 60)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 34)(21 35)(22 36)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 78)(30 79)(31 80)(32 81)(33 82)(45 74)(46 75)(47 76)(48 77)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 45)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 23)(20 24)(21 25)(22 26)(34 84)(35 85)(36 86)(37 87)(38 88)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(56 70)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 67)(65 68)(66 69)
(12 37 27 87)(13 38 28 88)(14 39 29 78)(15 40 30 79)(16 41 31 80)(17 42 32 81)(18 43 33 82)(19 44 23 83)(20 34 24 84)(21 35 25 85)(22 36 26 86)(56 70)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 67)(65 68)(66 69)

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

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

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

121 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4E 11A ··· 11J 22A ··· 22J 22K ··· 22AN 22AO ··· 22AX 44A ··· 44AX order 1 2 2 2 2 2 4 ··· 4 11 ··· 11 22 ··· 22 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 2 2 2 4 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

121 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C4 C4 C11 C22 C22 C44 C44 D4 D4×C11 C23⋊C4 C11×C23⋊C4 kernel C11×C23⋊C4 C11×C22⋊C4 D4×C22 C2×C44 C22×C22 C23⋊C4 C22⋊C4 C2×D4 C2×C4 C23 C2×C22 C22 C11 C1 # reps 1 2 1 2 2 10 20 10 20 20 2 20 1 10

Matrix representation of C11×C23⋊C4 in GL4(𝔽89) generated by

 45 0 0 0 0 45 0 0 0 0 45 0 0 0 0 45
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 88 0 0 0 0 88 0 0 0 0 88 0 0 0 0 88
,
 1 0 0 0 0 88 0 0 0 0 0 88 0 0 1 0
G:=sub<GL(4,GF(89))| [45,0,0,0,0,45,0,0,0,0,45,0,0,0,0,45],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,88,0,0,0,0,0,1,0,0,88,0] >;

C11×C23⋊C4 in GAP, Magma, Sage, TeX

C_{11}\times C_2^3\rtimes C_4
% in TeX

G:=Group("C11xC2^3:C4");
// GroupNames label

G:=SmallGroup(352,48);
// by ID

G=gap.SmallGroup(352,48);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,5283,3970]);
// Polycyclic

G:=Group<a,b,c,d,e|a^11=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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