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G = C31⋊D4order 248 = 23·31

The semidirect product of C31 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C312D4, C22⋊D31, D622C2, Dic31⋊C2, C2.5D62, C62.5C22, (C2×C62)⋊2C2, SmallGroup(248,7)

Series: Derived Chief Lower central Upper central

C1C62 — C31⋊D4
C1C31C62D62 — C31⋊D4
C31C62 — C31⋊D4
C1C2C22

Generators and relations for C31⋊D4
 G = < a,b,c | a31=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
62C2
31C4
31C22
2D31
2C62
31D4

Smallest permutation representation of C31⋊D4
On 124 points
Generators in S124
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(1 69 55 112)(2 68 56 111)(3 67 57 110)(4 66 58 109)(5 65 59 108)(6 64 60 107)(7 63 61 106)(8 93 62 105)(9 92 32 104)(10 91 33 103)(11 90 34 102)(12 89 35 101)(13 88 36 100)(14 87 37 99)(15 86 38 98)(16 85 39 97)(17 84 40 96)(18 83 41 95)(19 82 42 94)(20 81 43 124)(21 80 44 123)(22 79 45 122)(23 78 46 121)(24 77 47 120)(25 76 48 119)(26 75 49 118)(27 74 50 117)(28 73 51 116)(29 72 52 115)(30 71 53 114)(31 70 54 113)
(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 124)(89 123)(90 122)(91 121)(92 120)(93 119)

G:=sub<Sym(124)| (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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,69,55,112)(2,68,56,111)(3,67,57,110)(4,66,58,109)(5,65,59,108)(6,64,60,107)(7,63,61,106)(8,93,62,105)(9,92,32,104)(10,91,33,103)(11,90,34,102)(12,89,35,101)(13,88,36,100)(14,87,37,99)(15,86,38,98)(16,85,39,97)(17,84,40,96)(18,83,41,95)(19,82,42,94)(20,81,43,124)(21,80,44,123)(22,79,45,122)(23,78,46,121)(24,77,47,120)(25,76,48,119)(26,75,49,118)(27,74,50,117)(28,73,51,116)(29,72,52,115)(30,71,53,114)(31,70,54,113), (2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119)>;

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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,69,55,112)(2,68,56,111)(3,67,57,110)(4,66,58,109)(5,65,59,108)(6,64,60,107)(7,63,61,106)(8,93,62,105)(9,92,32,104)(10,91,33,103)(11,90,34,102)(12,89,35,101)(13,88,36,100)(14,87,37,99)(15,86,38,98)(16,85,39,97)(17,84,40,96)(18,83,41,95)(19,82,42,94)(20,81,43,124)(21,80,44,123)(22,79,45,122)(23,78,46,121)(24,77,47,120)(25,76,48,119)(26,75,49,118)(27,74,50,117)(28,73,51,116)(29,72,52,115)(30,71,53,114)(31,70,54,113), (2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119) );

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),(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93),(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,69,55,112),(2,68,56,111),(3,67,57,110),(4,66,58,109),(5,65,59,108),(6,64,60,107),(7,63,61,106),(8,93,62,105),(9,92,32,104),(10,91,33,103),(11,90,34,102),(12,89,35,101),(13,88,36,100),(14,87,37,99),(15,86,38,98),(16,85,39,97),(17,84,40,96),(18,83,41,95),(19,82,42,94),(20,81,43,124),(21,80,44,123),(22,79,45,122),(23,78,46,121),(24,77,47,120),(25,76,48,119),(26,75,49,118),(27,74,50,117),(28,73,51,116),(29,72,52,115),(30,71,53,114),(31,70,54,113)], [(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,124),(89,123),(90,122),(91,121),(92,120),(93,119)])

C31⋊D4 is a maximal subgroup of   D1245C2  D4×D31  D42D31
C31⋊D4 is a maximal quotient of   Dic31⋊C4  D62⋊C4  D4⋊D31  D4.D31  Q8⋊D31  C31⋊Q16  C23.D31

65 conjugacy classes

class 1 2A2B2C 4 31A···31O62A···62AS
order1222431···3162···62
size11262622···22···2

65 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D31D62C31⋊D4
kernelC31⋊D4Dic31D62C2×C62C31C22C2C1
# reps11111151530

Matrix representation of C31⋊D4 in GL2(𝔽373) generated by

301
275233
,
100177
346273
,
13289
2360
G:=sub<GL(2,GF(373))| [30,275,1,233],[100,346,177,273],[13,2,289,360] >;

C31⋊D4 in GAP, Magma, Sage, TeX

C_{31}\rtimes D_4
% in TeX

G:=Group("C31:D4");
// GroupNames label

G:=SmallGroup(248,7);
// by ID

G=gap.SmallGroup(248,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-31,49,3843]);
// Polycyclic

G:=Group<a,b,c|a^31=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C31⋊D4 in TeX

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