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G = D4:6D22order 352 = 25·11

2nd semidirect product of D4 and D22 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:6D22, C23:2D22, D44:8C22, C22.7C24, C44.21C23, D22.3C23, C11:12+ 1+4, Dic22:8C22, Dic11.4C23, (C2xC4):3D22, (D4xC22):7C2, (C2xD4):7D11, (D4xD11):4C2, (C2xC44):3C22, D44:5C2:5C2, D4:2D11:4C2, (C4xD11):1C22, (D4xC11):7C22, C11:D4:3C22, (C2xC22).2C23, C2.8(C23xD11), (C22xC22):5C22, C4.21(C22xD11), (C2xDic11):4C22, (C22xD11):3C22, C22.6(C22xD11), (C2xC11:D4):11C2, SmallGroup(352,179)

Series: Derived Chief Lower central Upper central

C1C22 — D4:6D22
C1C11C22D22C22xD11D4xD11 — D4:6D22
C11C22 — D4:6D22
C1C2C2xD4

Generators and relations for D4:6D22
 G = < a,b,c,d | a4=b2=c22=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 970 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C11, C2xD4, C2xD4, C4oD4, D11, C22, C22, 2+ 1+4, Dic11, C44, D22, D22, C2xC22, C2xC22, C2xC22, Dic22, C4xD11, D44, C2xDic11, C11:D4, C2xC44, D4xC11, C22xD11, C22xC22, D44:5C2, D4xD11, D4:2D11, C2xC11:D4, D4xC22, D4:6D22
Quotients: C1, C2, C22, C23, C24, D11, 2+ 1+4, D22, C22xD11, C23xD11, D4:6D22

Smallest permutation representation of D4:6D22
On 88 points
Generators in S88
(1 77 20 88)(2 67 21 78)(3 79 22 68)(4 69 12 80)(5 81 13 70)(6 71 14 82)(7 83 15 72)(8 73 16 84)(9 85 17 74)(10 75 18 86)(11 87 19 76)(23 60 34 49)(24 50 35 61)(25 62 36 51)(26 52 37 63)(27 64 38 53)(28 54 39 65)(29 66 40 55)(30 56 41 45)(31 46 42 57)(32 58 43 47)(33 48 44 59)
(1 57)(2 47)(3 59)(4 49)(5 61)(6 51)(7 63)(8 53)(9 65)(10 55)(11 45)(12 60)(13 50)(14 62)(15 52)(16 64)(17 54)(18 66)(19 56)(20 46)(21 58)(22 48)(23 80)(24 70)(25 82)(26 72)(27 84)(28 74)(29 86)(30 76)(31 88)(32 78)(33 68)(34 69)(35 81)(36 71)(37 83)(38 73)(39 85)(40 75)(41 87)(42 77)(43 67)(44 79)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 16)(13 15)(17 22)(18 21)(19 20)(23 38)(24 37)(25 36)(26 35)(27 34)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(45 46)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(67 75)(68 74)(69 73)(70 72)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)

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

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

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

67 conjugacy classes

class 1 2A2B···2F2G2H2I2J4A4B4C4D4E4F11A···11E22A···22O22P···22AI44A···44J
order122···2222244444411···1122···2222···2244···44
size112···22222222222222222222···22···24···44···4

67 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2D11D22D22D222+ 1+4D4:6D22
kernelD4:6D22D44:5C2D4xD11D4:2D11C2xC11:D4D4xC22C2xD4C2xC4D4C23C11C1
# reps124441552010110

Matrix representation of D4:6D22 in GL4(F89) generated by

4728851
61425054
7024323
1182846
,
88000
08800
59310
636801
,
2134152
5534214
001247
003422
,
34218410
34558775
001782
00372
G:=sub<GL(4,GF(89))| [47,61,70,11,28,42,2,8,85,50,43,28,1,54,23,46],[88,0,59,63,0,88,3,68,0,0,1,0,0,0,0,1],[21,55,0,0,34,34,0,0,1,2,12,34,52,14,47,22],[34,34,0,0,21,55,0,0,84,87,17,3,10,75,82,72] >;

D4:6D22 in GAP, Magma, Sage, TeX

D_4\rtimes_6D_{22}
% in TeX

G:=Group("D4:6D22");
// GroupNames label

G:=SmallGroup(352,179);
// by ID

G=gap.SmallGroup(352,179);
# by ID

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

G:=Group<a,b,c,d|a^4=b^2=c^22=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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