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G = C2xD4xD11order 352 = 25·11

Direct product of C2, D4 and D11

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

Aliases: C2xD4xD11, C44:C23, C23:3D22, D44:7C22, D22:2C23, C22.5C24, Dic11:1C23, C22:2(C2xD4), (C2xC4):6D22, (C2xC22):C23, (D4xC22):5C2, C11:2(C22xD4), (C2xD44):11C2, (C2xC44):2C22, C4:1(C22xD11), (C23xD11):4C2, (D4xC11):5C22, (C4xD11):3C22, C11:D4:1C22, C2.6(C23xD11), (C22xC22):4C22, C22:1(C22xD11), (C2xDic11):8C22, (C22xD11):6C22, (C2xC4xD11):3C2, (C2xC11:D4):9C2, SmallGroup(352,177)

Series: Derived Chief Lower central Upper central

C1C22 — C2xD4xD11
C1C11C22D22C22xD11C23xD11 — C2xD4xD11
C11C22 — C2xD4xD11
C1C22C2xD4

Generators and relations for C2xD4xD11
 G = < a,b,c,d,e | a2=b4=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1546 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, D4, C23, C23, C11, C22xC4, C2xD4, C2xD4, C24, D11, D11, C22, C22, C22, C22xD4, Dic11, C44, D22, D22, C2xC22, C2xC22, C2xC22, C4xD11, D44, C2xDic11, C11:D4, C2xC44, D4xC11, C22xD11, C22xD11, C22xD11, C22xC22, C2xC4xD11, C2xD44, D4xD11, C2xC11:D4, D4xC22, C23xD11, C2xD4xD11
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, D11, C22xD4, D22, C22xD11, D4xD11, C23xD11, C2xD4xD11

Smallest permutation representation of C2xD4xD11
On 88 points
Generators in S88
(1 32)(2 33)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(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 54 21 65)(2 55 22 66)(3 45 12 56)(4 46 13 57)(5 47 14 58)(6 48 15 59)(7 49 16 60)(8 50 17 61)(9 51 18 62)(10 52 19 63)(11 53 20 64)(23 67 34 78)(24 68 35 79)(25 69 36 80)(26 70 37 81)(27 71 38 82)(28 72 39 83)(29 73 40 84)(30 74 41 85)(31 75 42 86)(32 76 43 87)(33 77 44 88)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 33)(20 32)(21 31)(22 30)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 88)(53 87)(54 86)(55 85)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 77)(64 76)(65 75)(66 74)

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

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

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

70 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D11A···11E22A···22O22P···22AI44A···44J
order1222222222222222444411···1122···2222···2244···44
size1111222211111111222222222222222···22···24···44···4

70 irreducible representations

dim1111111222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D11D22D22D22D4xD11
kernelC2xD4xD11C2xC4xD11C2xD44D4xD11C2xC11:D4D4xC22C23xD11D22C2xD4C2xC4D4C23C2
# reps1118212455201010

Matrix representation of C2xD4xD11 in GL5(F89)

880000
01000
00100
000880
000088
,
10000
088000
008800
000088
00010
,
10000
01000
00100
00010
000088
,
10000
042100
0425400
00010
00001
,
10000
0548800
0673500
00010
00001

G:=sub<GL(5,GF(89))| [88,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,0,0,0,0,0,88],[1,0,0,0,0,0,88,0,0,0,0,0,88,0,0,0,0,0,0,1,0,0,0,88,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88],[1,0,0,0,0,0,42,42,0,0,0,1,54,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,54,67,0,0,0,88,35,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2xD4xD11 in GAP, Magma, Sage, TeX

C_2\times D_4\times D_{11}
% in TeX

G:=Group("C2xD4xD11");
// GroupNames label

G:=SmallGroup(352,177);
// by ID

G=gap.SmallGroup(352,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,159,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^11=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,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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