Copied to
clipboard

## G = C22⋊C4×D13order 416 = 25·13

### Direct product of C22⋊C4 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C22⋊C4×D13
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C23×D13 — C22⋊C4×D13
 Lower central C13 — C26 — C22⋊C4×D13
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22⋊C4×D13
G = < a,b,c,d,e | a2=b2=c4=d13=e2=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1040 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C13, C22⋊C4, C22⋊C4, C22×C4, C24, D13, D13, C26, C26, C26, C2×C22⋊C4, Dic13, C52, D26, D26, C2×C26, C2×C26, C2×C26, C4×D13, C2×Dic13, C2×C52, C22×D13, C22×D13, C22×D13, C22×C26, D26⋊C4, C23.D13, C13×C22⋊C4, C2×C4×D13, C23×D13, C22⋊C4×D13
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, D13, C2×C22⋊C4, D26, C4×D13, C22×D13, C2×C4×D13, D4×D13, C22⋊C4×D13

Smallest permutation representation of C22⋊C4×D13
On 104 points
Generators in S104
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 27)(14 51)(15 52)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(61 104)(62 92)(63 93)(64 94)(65 95)(66 84)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 91)(74 79)(75 80)(76 81)(77 82)(78 83)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 14)(9 15)(10 16)(11 17)(12 18)(13 19)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 40)(38 41)(39 42)(53 75)(54 76)(55 77)(56 78)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 92)(90 93)(91 94)
(1 86 28 59)(2 87 29 60)(3 88 30 61)(4 89 31 62)(5 90 32 63)(6 91 33 64)(7 79 34 65)(8 80 35 53)(9 81 36 54)(10 82 37 55)(11 83 38 56)(12 84 39 57)(13 85 27 58)(14 96 51 75)(15 97 52 76)(16 98 40 77)(17 99 41 78)(18 100 42 66)(19 101 43 67)(20 102 44 68)(21 103 45 69)(22 104 46 70)(23 92 47 71)(24 93 48 72)(25 94 49 73)(26 95 50 74)
(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 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 43)(2 42)(3 41)(4 40)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(53 94)(54 93)(55 92)(56 104)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 91)(76 90)(77 89)(78 88)

G:=sub<Sym(104)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,27)(14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(61,104)(62,92)(63,93)(64,94)(65,95)(66,84)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,79)(75,80)(76,81)(77,82)(78,83), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,14)(9,15)(10,16)(11,17)(12,18)(13,19)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,40)(38,41)(39,42)(53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,92)(90,93)(91,94), (1,86,28,59)(2,87,29,60)(3,88,30,61)(4,89,31,62)(5,90,32,63)(6,91,33,64)(7,79,34,65)(8,80,35,53)(9,81,36,54)(10,82,37,55)(11,83,38,56)(12,84,39,57)(13,85,27,58)(14,96,51,75)(15,97,52,76)(16,98,40,77)(17,99,41,78)(18,100,42,66)(19,101,43,67)(20,102,44,68)(21,103,45,69)(22,104,46,70)(23,92,47,71)(24,93,48,72)(25,94,49,73)(26,95,50,74), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,43)(2,42)(3,41)(4,40)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(53,94)(54,93)(55,92)(56,104)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,91)(76,90)(77,89)(78,88)>;

G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,27)(14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(61,104)(62,92)(63,93)(64,94)(65,95)(66,84)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,79)(75,80)(76,81)(77,82)(78,83), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,14)(9,15)(10,16)(11,17)(12,18)(13,19)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,40)(38,41)(39,42)(53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,92)(90,93)(91,94), (1,86,28,59)(2,87,29,60)(3,88,30,61)(4,89,31,62)(5,90,32,63)(6,91,33,64)(7,79,34,65)(8,80,35,53)(9,81,36,54)(10,82,37,55)(11,83,38,56)(12,84,39,57)(13,85,27,58)(14,96,51,75)(15,97,52,76)(16,98,40,77)(17,99,41,78)(18,100,42,66)(19,101,43,67)(20,102,44,68)(21,103,45,69)(22,104,46,70)(23,92,47,71)(24,93,48,72)(25,94,49,73)(26,95,50,74), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,43)(2,42)(3,41)(4,40)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(53,94)(54,93)(55,92)(56,104)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,91)(76,90)(77,89)(78,88) );

G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,27),(14,51),(15,52),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(53,96),(54,97),(55,98),(56,99),(57,100),(58,101),(59,102),(60,103),(61,104),(62,92),(63,93),(64,94),(65,95),(66,84),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90),(73,91),(74,79),(75,80),(76,81),(77,82),(78,83)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,14),(9,15),(10,16),(11,17),(12,18),(13,19),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,40),(38,41),(39,42),(53,75),(54,76),(55,77),(56,78),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,73),(65,74),(79,95),(80,96),(81,97),(82,98),(83,99),(84,100),(85,101),(86,102),(87,103),(88,104),(89,92),(90,93),(91,94)], [(1,86,28,59),(2,87,29,60),(3,88,30,61),(4,89,31,62),(5,90,32,63),(6,91,33,64),(7,79,34,65),(8,80,35,53),(9,81,36,54),(10,82,37,55),(11,83,38,56),(12,84,39,57),(13,85,27,58),(14,96,51,75),(15,97,52,76),(16,98,40,77),(17,99,41,78),(18,100,42,66),(19,101,43,67),(20,102,44,68),(21,103,45,69),(22,104,46,70),(23,92,47,71),(24,93,48,72),(25,94,49,73),(26,95,50,74)], [(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),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,43),(2,42),(3,41),(4,40),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(53,94),(54,93),(55,92),(56,104),(57,103),(58,102),(59,101),(60,100),(61,99),(62,98),(63,97),(64,96),(65,95),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,91),(76,90),(77,89),(78,88)]])

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 13A ··· 13F 26A ··· 26R 26S ··· 26AD 52A ··· 52X order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 13 13 13 13 26 26 2 2 2 2 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D13 D26 D26 C4×D13 D4×D13 kernel C22⋊C4×D13 D26⋊C4 C23.D13 C13×C22⋊C4 C2×C4×D13 C23×D13 C22×D13 D26 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 1 1 2 1 8 4 6 12 6 24 12

Matrix representation of C22⋊C4×D13 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 1 0 0 0 49 52
,
 1 0 0 0 0 1 0 0 0 0 52 0 0 0 0 52
,
 30 0 0 0 0 30 0 0 0 0 49 51 0 0 34 4
,
 35 51 0 0 52 44 0 0 0 0 1 0 0 0 0 1
,
 18 48 0 0 1 35 0 0 0 0 52 0 0 0 0 52
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,1,49,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[30,0,0,0,0,30,0,0,0,0,49,34,0,0,51,4],[35,52,0,0,51,44,0,0,0,0,1,0,0,0,0,1],[18,1,0,0,48,35,0,0,0,0,52,0,0,0,0,52] >;

C22⋊C4×D13 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times D_{13}
% in TeX

G:=Group("C2^2:C4xD13");
// GroupNames label

G:=SmallGroup(416,101);
// by ID

G=gap.SmallGroup(416,101);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,188,50,13829]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^13=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽