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## G = C2×D44order 176 = 24·11

### Direct product of C2 and D44

Aliases: C2×D44, C42D22, C221D4, C442C22, D221C22, C22.3C23, C22.10D22, C111(C2×D4), (C2×C44)⋊3C2, (C2×C4)⋊2D11, (C22×D11)⋊1C2, C2.4(C22×D11), (C2×C22).10C22, SmallGroup(176,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×D44
 Chief series C1 — C11 — C22 — D22 — C22×D11 — C2×D44
 Lower central C11 — C22 — C2×D44
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D44
G = < a,b,c | a2=b44=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 340 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C11, C2×D4, D11, C22, C22, C44, D22, D22, C2×C22, D44, C2×C44, C22×D11, C2×D44
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, D22, D44, C22×D11, C2×D44

Smallest permutation representation of C2×D44
On 88 points
Generators in S88
(1 87)(2 88)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(85 88)(86 87)

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

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

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

C2×D44 is a maximal subgroup of
C22.D8  C2.D88  C44.46D4  C4⋊D44  C4.D44  C22⋊D44  D22⋊D4  D44⋊C4  D22.5D4  C42D44  C8⋊D22  C447D4  C44⋊D4  C44.23D4  Q8⋊D22  C2×D4×D11  D48D22
C2×D44 is a maximal quotient of
C442Q8  C4⋊D44  C4.D44  C22⋊D44  C22.D44  C42D44  D222Q8  D887C2  C8⋊D22  C8.D22  C447D4

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 11A ··· 11E 22A ··· 22O 44A ··· 44T order 1 2 2 2 2 2 2 2 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 22 22 22 22 2 2 2 ··· 2 2 ··· 2 2 ··· 2

50 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D11 D22 D22 D44 kernel C2×D44 D44 C2×C44 C22×D11 C22 C2×C4 C4 C22 C2 # reps 1 4 1 2 2 5 10 5 20

Matrix representation of C2×D44 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 88 0 0 0 0 88
,
 44 17 0 0 72 42 0 0 0 0 52 54 0 0 60 88
,
 44 17 0 0 38 45 0 0 0 0 8 74 0 0 22 81
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[44,72,0,0,17,42,0,0,0,0,52,60,0,0,54,88],[44,38,0,0,17,45,0,0,0,0,8,22,0,0,74,81] >;

C2×D44 in GAP, Magma, Sage, TeX

C_2\times D_{44}
% in TeX

G:=Group("C2xD44");
// GroupNames label

G:=SmallGroup(176,29);
// by ID

G=gap.SmallGroup(176,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,182,42,4004]);
// Polycyclic

G:=Group<a,b,c|a^2=b^44=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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