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## G = C11×C4○D4order 176 = 24·11

### Direct product of C11 and C4○D4

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

Aliases: C11×C4○D4, D42C22, Q82C22, C22.13C23, C44.21C22, (C2×C44)⋊7C2, (C2×C4)⋊3C22, C44(D4×C11), C44(Q8×C11), (D4×C11)⋊5C2, C4.5(C2×C22), (Q8×C11)⋊5C2, C22.(C2×C22), (C2×C22).2C22, C2.3(C22×C22), SmallGroup(176,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×C4○D4
 Chief series C1 — C2 — C22 — C2×C22 — D4×C11 — C11×C4○D4
 Lower central C1 — C2 — C11×C4○D4
 Upper central C1 — C44 — C11×C4○D4

Generators and relations for C11×C4○D4
G = < a,b,c,d | a11=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C11×C4○D4
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 53 31 38)(2 54 32 39)(3 55 33 40)(4 45 23 41)(5 46 24 42)(6 47 25 43)(7 48 26 44)(8 49 27 34)(9 50 28 35)(10 51 29 36)(11 52 30 37)(12 67 85 63)(13 68 86 64)(14 69 87 65)(15 70 88 66)(16 71 78 56)(17 72 79 57)(18 73 80 58)(19 74 81 59)(20 75 82 60)(21 76 83 61)(22 77 84 62)
(1 38 31 53)(2 39 32 54)(3 40 33 55)(4 41 23 45)(5 42 24 46)(6 43 25 47)(7 44 26 48)(8 34 27 49)(9 35 28 50)(10 36 29 51)(11 37 30 52)(12 67 85 63)(13 68 86 64)(14 69 87 65)(15 70 88 66)(16 71 78 56)(17 72 79 57)(18 73 80 58)(19 74 81 59)(20 75 82 60)(21 76 83 61)(22 77 84 62)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 78)(9 79)(10 80)(11 81)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 67)(42 68)(43 69)(44 70)(45 63)(46 64)(47 65)(48 66)(49 56)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)

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

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

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

C11×C4○D4 is a maximal subgroup of   C44.56D4  Q8.Dic11  Q8⋊D22  D4.8D22  D4.9D22  D48D22  D4.10D22
C11×C4○D4 is a maximal quotient of   D4×C44  Q8×C44

110 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 11A ··· 11J 22A ··· 22J 22K ··· 22AN 44A ··· 44T 44U ··· 44AX order 1 2 2 2 2 4 4 4 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 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 C4○D4 C11×C4○D4 kernel C11×C4○D4 C2×C44 D4×C11 Q8×C11 C4○D4 C2×C4 D4 Q8 C11 C1 # reps 1 3 3 1 10 30 30 10 2 20

Matrix representation of C11×C4○D4 in GL2(𝔽89) generated by

 64 0 0 64
,
 55 0 0 55
,
 34 0 0 55
,
 0 55 34 0
G:=sub<GL(2,GF(89))| [64,0,0,64],[55,0,0,55],[34,0,0,55],[0,34,55,0] >;

C11×C4○D4 in GAP, Magma, Sage, TeX

C_{11}\times C_4\circ D_4
% in TeX

G:=Group("C11xC4oD4");
// GroupNames label

G:=SmallGroup(176,40);
// by ID

G=gap.SmallGroup(176,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-11,-2,901,342]);
// Polycyclic

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

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