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G = C2xD44order 176 = 24·11

Direct product of C2 and D44

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

Aliases: C2xD44, C4:2D22, C22:1D4, C44:2C22, D22:1C22, C22.3C23, C22.10D22, C11:1(C2xD4), (C2xC44):3C2, (C2xC4):2D11, (C22xD11):1C2, C2.4(C22xD11), (C2xC22).10C22, SmallGroup(176,29)

Series: Derived Chief Lower central Upper central

C1C22 — C2xD44
C1C11C22D22C22xD11 — C2xD44
C11C22 — C2xD44
C1C22C2xC4

Generators and relations for C2xD44
 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, C2xC4, D4, C23, C11, C2xD4, D11, C22, C22, C44, D22, D22, C2xC22, D44, C2xC44, C22xD11, C2xD44
Quotients: C1, C2, C22, D4, C23, C2xD4, D11, D22, D44, C22xD11, C2xD44

Smallest permutation representation of C2xD44
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)]])

C2xD44 is a maximal subgroup of
C22.D8  C2.D88  C44.46D4  C4:D44  C4.D44  C22:D44  D22:D4  D44:C4  D22.5D4  C4:2D44  C8:D22  C44:7D4  C44:D4  C44.23D4  Q8:D22  C2xD4xD11  D4:8D22
C2xD44 is a maximal quotient of
C44:2Q8  C4:D44  C4.D44  C22:D44  C22.D44  C4:2D44  D22:2Q8  D88:7C2  C8:D22  C8.D22  C44:7D4

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B11A···11E22A···22O44A···44T
order122222224411···1122···2244···44
size111122222222222···22···22···2

50 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D11D22D22D44
kernelC2xD44D44C2xC44C22xD11C22C2xC4C4C22C2
# reps14122510520

Matrix representation of C2xD44 in GL4(F89) generated by

88000
08800
00880
00088
,
441700
724200
005254
006088
,
441700
384500
00874
002281
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] >;

C2xD44 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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