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G = C23xD11order 176 = 24·11

Direct product of C23 and D11

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

Aliases: C23xD11, C11:C24, C22:C23, (C22xC22):3C2, (C2xC22):4C22, SmallGroup(176,41)

Series: Derived Chief Lower central Upper central

C1C11 — C23xD11
C1C11D11D22C22xD11 — C23xD11
C11 — C23xD11
C1C23

Generators and relations for C23xD11
 G = < a,b,c,d,e | a2=b2=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 644 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C22, C22, C23, C23, C11, C24, D11, C22, D22, C2xC22, C22xD11, C22xC22, C23xD11
Quotients: C1, C2, C22, C23, C24, D11, D22, C22xD11, C23xD11

Smallest permutation representation of C23xD11
On 88 points
Generators in S88
(1 54)(2 55)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(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)
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 55)(11 54)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 66)(20 65)(21 64)(22 63)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 77)(31 76)(32 75)(33 74)(34 84)(35 83)(36 82)(37 81)(38 80)(39 79)(40 78)(41 88)(42 87)(43 86)(44 85)

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

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

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

C23xD11 is a maximal subgroup of   C22:D44  C23:D22
C23xD11 is a maximal quotient of   D4:6D22  Q8.10D22  D4:8D22  D4.10D22

56 conjugacy classes

class 1 2A···2G2H···2O11A···11E22A···22AI
order12···22···211···1122···22
size11···111···112···22···2

56 irreducible representations

dim11122
type+++++
imageC1C2C2D11D22
kernelC23xD11C22xD11C22xC22C23C22
# reps1141535

Matrix representation of C23xD11 in GL4(F23) generated by

22000
0100
0010
0001
,
1000
02200
0010
0001
,
22000
02200
00220
00022
,
1000
0100
0001
002210
,
1000
02200
0001
0010
G:=sub<GL(4,GF(23))| [22,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,22,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,22,0,0,0,0,22,0,0,0,0,22],[1,0,0,0,0,1,0,0,0,0,0,22,0,0,1,10],[1,0,0,0,0,22,0,0,0,0,0,1,0,0,1,0] >;

C23xD11 in GAP, Magma, Sage, TeX

C_2^3\times D_{11}
% in TeX

G:=Group("C2^3xD11");
// GroupNames label

G:=SmallGroup(176,41);
// by ID

G=gap.SmallGroup(176,41);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,4004]);
// Polycyclic

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

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