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## G = C22×D31order 248 = 23·31

### Direct product of C22 and D31

Aliases: C22×D31, C31⋊C23, C62⋊C22, (C2×C62)⋊3C2, SmallGroup(248,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — C22×D31
 Chief series C1 — C31 — D31 — D62 — C22×D31
 Lower central C31 — C22×D31
 Upper central C1 — C22

Generators and relations for C22×D31
G = < a,b,c,d | a2=b2=c31=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C22×D31
On 124 points
Generators in S124
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 94)(31 95)(32 89)(33 90)(34 91)(35 92)(36 93)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 32)(28 33)(29 34)(30 35)(31 36)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 121)(89 122)(90 123)(91 124)(92 94)(93 95)
(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 95)(2 94)(3 124)(4 123)(5 122)(6 121)(7 120)(8 119)(9 118)(10 117)(11 116)(12 115)(13 114)(14 113)(15 112)(16 111)(17 110)(18 109)(19 108)(20 107)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 100)(28 99)(29 98)(30 97)(31 96)(32 67)(33 66)(34 65)(35 64)(36 63)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)

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

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

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

C22×D31 is a maximal subgroup of   D62⋊C4
C22×D31 is a maximal quotient of   D1245C2  D42D31  Q82D31

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 31A ··· 31O 62A ··· 62AS order 1 2 2 2 2 2 2 2 31 ··· 31 62 ··· 62 size 1 1 1 1 31 31 31 31 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D31 D62 kernel C22×D31 D62 C2×C62 C22 C2 # reps 1 6 1 15 45

Matrix representation of C22×D31 in GL3(𝔽311) generated by

 1 0 0 0 310 0 0 0 310
,
 310 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 310 50
,
 310 0 0 0 0 310 0 310 0
G:=sub<GL(3,GF(311))| [1,0,0,0,310,0,0,0,310],[310,0,0,0,1,0,0,0,1],[1,0,0,0,0,310,0,1,50],[310,0,0,0,0,310,0,310,0] >;

C22×D31 in GAP, Magma, Sage, TeX

C_2^2\times D_{31}
% in TeX

G:=Group("C2^2xD31");
// GroupNames label

G:=SmallGroup(248,11);
// by ID

G=gap.SmallGroup(248,11);
# by ID

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

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

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