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## G = C22×C62order 144 = 24·32

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

Aliases: C22×C62, SmallGroup(144,197)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C62
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C62 — C22×C62
 Lower central C1 — C22×C62
 Upper central 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 ··· 2O 3A ··· 3H 6A ··· 6DP order 1 2 ··· 2 3 ··· 3 6 ··· 6 size 1 1 ··· 1 1 ··· 1 1 ··· 1

144 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C3 C6 kernel C22×C62 C2×C62 C23×C6 C22×C6 # reps 1 15 8 120

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

 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
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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