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## G = D4×C22order 176 = 24·11

### Direct product of C22 and D4

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

Aliases: D4×C22, C23⋊C22, C444C22, C22.11C23, C4⋊(C2×C22), (C2×C44)⋊6C2, (C2×C4)⋊2C22, C22⋊(C2×C22), (C22×C22)⋊1C2, (C2×C22)⋊2C22, C2.1(C22×C22), SmallGroup(176,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C22
 Chief series C1 — C2 — C22 — C2×C22 — D4×C11 — D4×C22
 Lower central C1 — C2 — D4×C22
 Upper central C1 — C2×C22 — D4×C22

Generators and relations for D4×C22
G = < a,b,c | a22=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C11, C2×D4, C22, C22, C22, C44, C2×C22, C2×C22, C2×C22, C2×C44, D4×C11, C22×C22, D4×C22
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C22, C2×C22, D4×C11, C22×C22, D4×C22

Smallest permutation representation of D4×C22
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 29 53 87)(2 30 54 88)(3 31 55 67)(4 32 56 68)(5 33 57 69)(6 34 58 70)(7 35 59 71)(8 36 60 72)(9 37 61 73)(10 38 62 74)(11 39 63 75)(12 40 64 76)(13 41 65 77)(14 42 66 78)(15 43 45 79)(16 44 46 80)(17 23 47 81)(18 24 48 82)(19 25 49 83)(20 26 50 84)(21 27 51 85)(22 28 52 86)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 67)

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

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

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

D4×C22 is a maximal subgroup of
D4⋊Dic11  C44.D4  C23⋊Dic11  D446C22  C23.18D22  C44.17D4  C23⋊D22  C442D4  Dic11⋊D4  C44⋊D4  D46D22

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 11A ··· 11J 22A ··· 22AD 22AE ··· 22BR 44A ··· 44T order 1 2 2 2 2 2 2 2 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C11 C22 C22 C22 D4 D4×C11 kernel D4×C22 C2×C44 D4×C11 C22×C22 C2×D4 C2×C4 D4 C23 C22 C2 # reps 1 1 4 2 10 10 40 20 2 20

Matrix representation of D4×C22 in GL3(𝔽89) generated by

 88 0 0 0 81 0 0 0 81
,
 1 0 0 0 1 88 0 2 88
,
 1 0 0 0 88 0 0 87 1
G:=sub<GL(3,GF(89))| [88,0,0,0,81,0,0,0,81],[1,0,0,0,1,2,0,88,88],[1,0,0,0,88,87,0,0,1] >;

D4×C22 in GAP, Magma, Sage, TeX

D_4\times C_{22}
% in TeX

G:=Group("D4xC22");
// GroupNames label

G:=SmallGroup(176,38);
// by ID

G=gap.SmallGroup(176,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-11,-2,901]);
// Polycyclic

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

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