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## G = C22×C30order 120 = 23·3·5

### Abelian group of type [2,2,30]

Aliases: C22×C30, SmallGroup(120,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C30
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×C30
 Lower central C1 — C22×C30
 Upper central C1 — C22×C30

Generators and relations for C22×C30
G = < a,b,c | a2=b2=c30=1, ab=ba, ac=ca, bc=cb >

Subgroups: 64, all normal (8 characteristic)
C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], C23, C10 [×7], C2×C6 [×7], C15, C2×C10 [×7], C22×C6, C30 [×7], C22×C10, C2×C30 [×7], C22×C30
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], C23, C10 [×7], C2×C6 [×7], C15, C2×C10 [×7], C22×C6, C30 [×7], C22×C10, C2×C30 [×7], C22×C30

Smallest permutation representation of C22×C30
Regular action on 120 points
Generators in S120
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)
(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)

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

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

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

C22×C30 is a maximal subgroup of   C30.38D4

120 conjugacy classes

 class 1 2A ··· 2G 3A 3B 5A 5B 5C 5D 6A ··· 6N 10A ··· 10AB 15A ··· 15H 30A ··· 30BD order 1 2 ··· 2 3 3 5 5 5 5 6 ··· 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 ··· 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C5 C6 C10 C15 C30 kernel C22×C30 C2×C30 C22×C10 C22×C6 C2×C10 C2×C6 C23 C22 # reps 1 7 2 4 14 28 8 56

Matrix representation of C22×C30 in GL3(𝔽31) generated by

 30 0 0 0 1 0 0 0 1
,
 30 0 0 0 30 0 0 0 1
,
 30 0 0 0 26 0 0 0 17
G:=sub<GL(3,GF(31))| [30,0,0,0,1,0,0,0,1],[30,0,0,0,30,0,0,0,1],[30,0,0,0,26,0,0,0,17] >;

C22×C30 in GAP, Magma, Sage, TeX

C_2^2\times C_{30}
% in TeX

G:=Group("C2^2xC30");
// GroupNames label

G:=SmallGroup(120,47);
// by ID

G=gap.SmallGroup(120,47);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^30=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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