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G = D22⋊C4order 176 = 24·11

The semidirect product of D22 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D22⋊C4, C22.6D4, C2.2D44, C22.6D22, (C2×C44)⋊1C2, (C2×C4)⋊1D11, C22.5(C2×C4), C2.5(C4×D11), C111(C22⋊C4), (C22×D11).C2, (C2×Dic11)⋊1C2, C2.2(C11⋊D4), (C2×C22).6C22, SmallGroup(176,13)

Series: Derived Chief Lower central Upper central

C1C22 — D22⋊C4
C1C11C22C2×C22C22×D11 — D22⋊C4
C11C22 — D22⋊C4
C1C22C2×C4

Generators and relations for D22⋊C4
 G = < a,b,c | a22=b2=c4=1, bab=a-1, ac=ca, cbc-1=a11b >

22C2
22C2
2C4
11C22
11C22
22C4
22C22
22C22
2D11
2D11
11C2×C4
11C23
2D22
2D22
2Dic11
2C44
11C22⋊C4

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

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

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

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

D22⋊C4 is a maximal subgroup of
C42⋊D11  C4×D44  C4.D44  C422D11  C22⋊C4×D11  Dic114D4  C22⋊D44  D22.D4  D22⋊D4  Dic11.D4  C22.D44  C4⋊C47D11  D44⋊C4  D22.5D4  C42D44  D22⋊Q8  D222Q8  C4⋊C4⋊D11  C4×C11⋊D4  C23.23D22  C447D4  C23⋊D22  Dic11⋊D4  D223Q8  C44.23D4
D22⋊C4 is a maximal quotient of
D441C4  C22.2D44  C22.D8  C22.Q16  C44.44D4  D22⋊C8  C2.D88  C44.46D4  C44.47D4  D444C4  C22.C42

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D11A···11E22A···22O44A···44T
order122222444411···1122···2244···44
size111122222222222···22···22···2

50 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4D4D11D22C4×D11D44C11⋊D4
kernelD22⋊C4C2×Dic11C2×C44C22×D11D22C22C2×C4C22C2C2C2
# reps11114255101010

Matrix representation of D22⋊C4 in GL3(𝔽89) generated by

100
03630
08256
,
100
0062
0560
,
5500
0869
0783
G:=sub<GL(3,GF(89))| [1,0,0,0,36,82,0,30,56],[1,0,0,0,0,56,0,62,0],[55,0,0,0,86,78,0,9,3] >;

D22⋊C4 in GAP, Magma, Sage, TeX

D_{22}\rtimes C_4
% in TeX

G:=Group("D22:C4");
// GroupNames label

G:=SmallGroup(176,13);
// by ID

G=gap.SmallGroup(176,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,101,26,4004]);
// Polycyclic

G:=Group<a,b,c|a^22=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^11*b>;
// generators/relations

Export

Subgroup lattice of D22⋊C4 in TeX

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