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## G = C11×C22⋊C4order 176 = 24·11

### Direct product of C11 and C22⋊C4

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

Aliases: C11×C22⋊C4, C22⋊C44, C23.C22, C22.12D4, (C2×C22)⋊1C4, (C2×C4)⋊1C22, (C2×C44)⋊2C2, C2.1(C2×C44), C2.1(D4×C11), C22.10(C2×C4), (C22×C22).1C2, C22.2(C2×C22), (C2×C22).13C22, SmallGroup(176,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×C22⋊C4
 Chief series C1 — C2 — C22 — C2×C22 — C2×C44 — C11×C22⋊C4
 Lower central C1 — C2 — C11×C22⋊C4
 Upper central C1 — C2×C22 — C11×C22⋊C4

Generators and relations for C11×C22⋊C4
G = < a,b,c,d | a11=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C11×C22⋊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 73)(2 74)(3 75)(4 76)(5 77)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 42)(13 43)(14 44)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 78)(30 79)(31 80)(32 81)(33 82)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 56)(53 57)(54 58)(55 59)
(1 66)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 78)(21 79)(22 80)(23 44)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)
(1 30 73 21)(2 31 74 22)(3 32 75 12)(4 33 76 13)(5 23 77 14)(6 24 67 15)(7 25 68 16)(8 26 69 17)(9 27 70 18)(10 28 71 19)(11 29 72 20)(34 45 84 60)(35 46 85 61)(36 47 86 62)(37 48 87 63)(38 49 88 64)(39 50 78 65)(40 51 79 66)(41 52 80 56)(42 53 81 57)(43 54 82 58)(44 55 83 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), (1,73)(2,74)(3,75)(4,76)(5,77)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,42)(13,43)(14,44)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,78)(30,79)(31,80)(32,81)(33,82)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,56)(53,57)(54,58)(55,59), (1,66)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,78)(21,79)(22,80)(23,44)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77), (1,30,73,21)(2,31,74,22)(3,32,75,12)(4,33,76,13)(5,23,77,14)(6,24,67,15)(7,25,68,16)(8,26,69,17)(9,27,70,18)(10,28,71,19)(11,29,72,20)(34,45,84,60)(35,46,85,61)(36,47,86,62)(37,48,87,63)(38,49,88,64)(39,50,78,65)(40,51,79,66)(41,52,80,56)(42,53,81,57)(43,54,82,58)(44,55,83,59)>;

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

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

C11×C22⋊C4 is a maximal subgroup of
C22.2D44  C23.11D22  C22⋊Dic22  C23.D22  Dic114D4  C22⋊D44  D22.D4  D22⋊D4  Dic11.D4  C22.D44  D4×C44

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 11A ··· 11J 22A ··· 22AD 22AE ··· 22AX 44A ··· 44AN order 1 2 2 2 2 2 4 4 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 C4 C11 C22 C22 C44 D4 D4×C11 kernel C11×C22⋊C4 C2×C44 C22×C22 C2×C22 C22⋊C4 C2×C4 C23 C22 C22 C2 # reps 1 2 1 4 10 20 10 40 2 20

Matrix representation of C11×C22⋊C4 in GL3(𝔽89) generated by

 1 0 0 0 39 0 0 0 39
,
 1 0 0 0 1 25 0 0 88
,
 1 0 0 0 88 0 0 0 88
,
 55 0 0 0 25 45 0 87 64
G:=sub<GL(3,GF(89))| [1,0,0,0,39,0,0,0,39],[1,0,0,0,1,0,0,25,88],[1,0,0,0,88,0,0,0,88],[55,0,0,0,25,87,0,45,64] >;

C11×C22⋊C4 in GAP, Magma, Sage, TeX

C_{11}\times C_2^2\rtimes C_4
% in TeX

G:=Group("C11xC2^2:C4");
// GroupNames label

G:=SmallGroup(176,20);
// by ID

G=gap.SmallGroup(176,20);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-2,440,461]);
// Polycyclic

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

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