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## G = C2×D4×D11order 352 = 25·11

### Direct product of C2, D4 and D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×D4×D11
 Chief series C1 — C11 — C22 — D22 — C22×D11 — C23×D11 — C2×D4×D11
 Lower central C11 — C22 — C2×D4×D11
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×D11
G = < a,b,c,d,e | a2=b4=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1546 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, C23, C23, C11, C22×C4, C2×D4, C2×D4, C24, D11, D11, C22, C22, C22, C22×D4, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×C22, C4×D11, D44, C2×Dic11, C11⋊D4, C2×C44, D4×C11, C22×D11, C22×D11, C22×D11, C22×C22, C2×C4×D11, C2×D44, D4×D11, C2×C11⋊D4, D4×C22, C23×D11, C2×D4×D11
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, D11, C22×D4, D22, C22×D11, D4×D11, C23×D11, C2×D4×D11

Smallest permutation representation of C2×D4×D11
On 88 points
Generators in S88
(1 32)(2 33)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(1 54 21 65)(2 55 22 66)(3 45 12 56)(4 46 13 57)(5 47 14 58)(6 48 15 59)(7 49 16 60)(8 50 17 61)(9 51 18 62)(10 52 19 63)(11 53 20 64)(23 67 34 78)(24 68 35 79)(25 69 36 80)(26 70 37 81)(27 71 38 82)(28 72 39 83)(29 73 40 84)(30 74 41 85)(31 75 42 86)(32 76 43 87)(33 77 44 88)
(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 33)(20 32)(21 31)(22 30)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 88)(53 87)(54 86)(55 85)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 77)(64 76)(65 75)(66 74)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J order 1 2 2 2 2 2 2 2 2 2 2 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 11 11 11 11 22 22 22 22 2 2 22 22 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D11 D22 D22 D22 D4×D11 kernel C2×D4×D11 C2×C4×D11 C2×D44 D4×D11 C2×C11⋊D4 D4×C22 C23×D11 D22 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 1 8 2 1 2 4 5 5 20 10 10

Matrix representation of C2×D4×D11 in GL5(𝔽89)

 88 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 88 0 0 0 0 0 88
,
 1 0 0 0 0 0 88 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 88
,
 1 0 0 0 0 0 42 1 0 0 0 42 54 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 54 88 0 0 0 67 35 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(89))| [88,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,0,0,0,0,0,88],[1,0,0,0,0,0,88,0,0,0,0,0,88,0,0,0,0,0,0,1,0,0,0,88,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88],[1,0,0,0,0,0,42,42,0,0,0,1,54,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,54,67,0,0,0,88,35,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2×D4×D11 in GAP, Magma, Sage, TeX

C_2\times D_4\times D_{11}
% in TeX

G:=Group("C2xD4xD11");
// GroupNames label

G:=SmallGroup(352,177);
// by ID

G=gap.SmallGroup(352,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,159,11525]);
// Polycyclic

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

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