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G = C22×C62order 144 = 24·32

Abelian group of type [2,2,6,6]

direct product, abelian, monomial

Aliases: C22×C62, SmallGroup(144,197)

Series: Derived Chief Lower central Upper central

C1 — C22×C62
C1C3C32C3×C6C62C2×C62 — C22×C62
C1 — C22×C62
C1 — C22×C62

Generators and relations for C22×C62
 G = < a,b,c,d | a2=b2=c6=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 402, all normal (4 characteristic)
C1, C2 [×15], C3 [×4], C22 [×35], C6 [×60], C23 [×15], C32, C2×C6 [×140], C24, C3×C6 [×15], C22×C6 [×60], C62 [×35], C23×C6 [×4], C2×C62 [×15], C22×C62
Quotients: C1, C2 [×15], C3 [×4], C22 [×35], C6 [×60], C23 [×15], C32, C2×C6 [×140], C24, C3×C6 [×15], C22×C6 [×60], C62 [×35], C23×C6 [×4], C2×C62 [×15], C22×C62

Smallest permutation representation of C22×C62
Regular action on 144 points
Generators in S144
(1 82)(2 83)(3 84)(4 79)(5 80)(6 81)(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 87)(14 88)(15 89)(16 90)(17 85)(18 86)(19 99)(20 100)(21 101)(22 102)(23 97)(24 98)(25 95)(26 96)(27 91)(28 92)(29 93)(30 94)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 110)(8 111)(9 112)(10 113)(11 114)(12 109)(13 51)(14 52)(15 53)(16 54)(17 49)(18 50)(19 135)(20 136)(21 137)(22 138)(23 133)(24 134)(25 131)(26 132)(27 127)(28 128)(29 129)(30 130)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(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)(121 122 123 124 125 126)(127 128 129 130 131 132)(133 134 135 136 137 138)(139 140 141 142 143 144)
(1 32 58 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 117 139 29 121)(8 23 118 140 30 122)(9 24 119 141 25 123)(10 19 120 142 26 124)(11 20 115 143 27 125)(12 21 116 144 28 126)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)

G:=sub<Sym(144)| (1,82)(2,83)(3,84)(4,79)(5,80)(6,81)(7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,87)(14,88)(15,89)(16,90)(17,85)(18,86)(19,99)(20,100)(21,101)(22,102)(23,97)(24,98)(25,95)(26,96)(27,91)(28,92)(29,93)(30,94)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,110)(8,111)(9,112)(10,113)(11,114)(12,109)(13,51)(14,52)(15,53)(16,54)(17,49)(18,50)(19,135)(20,136)(21,137)(22,138)(23,133)(24,134)(25,131)(26,132)(27,127)(28,128)(29,129)(30,130)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (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)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135)>;

G:=Group( (1,82)(2,83)(3,84)(4,79)(5,80)(6,81)(7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,87)(14,88)(15,89)(16,90)(17,85)(18,86)(19,99)(20,100)(21,101)(22,102)(23,97)(24,98)(25,95)(26,96)(27,91)(28,92)(29,93)(30,94)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,110)(8,111)(9,112)(10,113)(11,114)(12,109)(13,51)(14,52)(15,53)(16,54)(17,49)(18,50)(19,135)(20,136)(21,137)(22,138)(23,133)(24,134)(25,131)(26,132)(27,127)(28,128)(29,129)(30,130)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (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)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135) );

G=PermutationGroup([(1,82),(2,83),(3,84),(4,79),(5,80),(6,81),(7,74),(8,75),(9,76),(10,77),(11,78),(12,73),(13,87),(14,88),(15,89),(16,90),(17,85),(18,86),(19,99),(20,100),(21,101),(22,102),(23,97),(24,98),(25,95),(26,96),(27,91),(28,92),(29,93),(30,94),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(1,46),(2,47),(3,48),(4,43),(5,44),(6,45),(7,110),(8,111),(9,112),(10,113),(11,114),(12,109),(13,51),(14,52),(15,53),(16,54),(17,49),(18,50),(19,135),(20,136),(21,137),(22,138),(23,133),(24,134),(25,131),(26,132),(27,127),(28,128),(29,129),(30,130),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,121),(86,122),(87,123),(88,124),(89,125),(90,126),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144)], [(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),(121,122,123,124,125,126),(127,128,129,130,131,132),(133,134,135,136,137,138),(139,140,141,142,143,144)], [(1,32,58,18,39,61),(2,33,59,13,40,62),(3,34,60,14,41,63),(4,35,55,15,42,64),(5,36,56,16,37,65),(6,31,57,17,38,66),(7,22,117,139,29,121),(8,23,118,140,30,122),(9,24,119,141,25,123),(10,19,120,142,26,124),(11,20,115,143,27,125),(12,21,116,144,28,126),(43,71,91,53,78,100),(44,72,92,54,73,101),(45,67,93,49,74,102),(46,68,94,50,75,97),(47,69,95,51,76,98),(48,70,96,52,77,99),(79,107,127,89,114,136),(80,108,128,90,109,137),(81,103,129,85,110,138),(82,104,130,86,111,133),(83,105,131,87,112,134),(84,106,132,88,113,135)])

C22×C62 is a maximal subgroup of   C6224D4  C62.A4  C62⋊A4

144 conjugacy classes

class 1 2A···2O3A···3H6A···6DP
order12···23···36···6
size11···11···11···1

144 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC22×C62C2×C62C23×C6C22×C6
# reps1158120

Matrix representation of C22×C62 in GL4(𝔽7) generated by

6000
0600
0060
0006
,
6000
0100
0060
0006
,
1000
0200
0050
0004
,
2000
0200
0020
0003
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[6,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,2,0,0,0,0,5,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,3] >;

C22×C62 in GAP, Magma, Sage, TeX

C_2^2\times C_6^2
% in TeX

G:=Group("C2^2xC6^2");
// GroupNames label

G:=SmallGroup(144,197);
// by ID

G=gap.SmallGroup(144,197);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^6=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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