direct product, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊C4×He3, C4⋊(C4×He3), C2.(Q8×He3), (C3×C12)⋊3C12, (C6×C12).3C6, (C4×He3)⋊7C4, C2.2(D4×He3), C6.24(C6×C12), C12.5(C3×C12), (C2×He3).8Q8, C6.4(Q8×C32), (C2×He3).40D4, (C2×C6).30C62, C62.35(C2×C6), C6.16(D4×C32), C22.4(C22×He3), (C22×He3).38C22, (C32×C4⋊C4)⋊C3, C2.4(C2×C4×He3), C32⋊8(C3×C4⋊C4), (C2×C4×He3).3C2, (C3×C6).6(C3×Q8), (C2×C12).5(C3×C6), (C3×C6).30(C3×D4), C3.2(C32×C4⋊C4), (C2×C4).1(C2×He3), (C3×C6).30(C2×C12), (C3×C4⋊C4).2C32, (C2×He3).37(C2×C4), SmallGroup(432,207)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×He3
G = < a,b,c,d,e | a4=b4=c3=d3=e3=1, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 285 in 143 conjugacy classes, 77 normal (21 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C32, C12, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×C12, C2×C12, C2×C12, He3, C3×C12, C3×C12, C62, C3×C4⋊C4, C3×C4⋊C4, C2×He3, C6×C12, C4×He3, C4×He3, C22×He3, C32×C4⋊C4, C2×C4×He3, C2×C4×He3, C4⋊C4×He3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4⋊C4, C3×C6, C2×C12, C3×D4, C3×Q8, He3, C3×C12, C62, C3×C4⋊C4, C2×He3, C6×C12, D4×C32, Q8×C32, C4×He3, C22×He3, C32×C4⋊C4, C2×C4×He3, D4×He3, Q8×He3, C4⋊C4×He3
►On 144 points
Generators in S144
(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)
(1 114 11 133)(2 113 12 136)(3 116 9 135)(4 115 10 134)(5 79 86 22)(6 78 87 21)(7 77 88 24)(8 80 85 23)(13 137 67 122)(14 140 68 121)(15 139 65 124)(16 138 66 123)(17 105 27 117)(18 108 28 120)(19 107 25 119)(20 106 26 118)(29 46 81 129)(30 45 82 132)(31 48 83 131)(32 47 84 130)(33 90 49 99)(34 89 50 98)(35 92 51 97)(36 91 52 100)(37 109 128 60)(38 112 125 59)(39 111 126 58)(40 110 127 57)(41 101 62 76)(42 104 63 75)(43 103 64 74)(44 102 61 73)(53 141 71 95)(54 144 72 94)(55 143 69 93)(56 142 70 96)
(5 131 127)(6 132 128)(7 129 125)(8 130 126)(13 53 36)(14 54 33)(15 55 34)(16 56 35)(21 82 109)(22 83 110)(23 84 111)(24 81 112)(29 59 77)(30 60 78)(31 57 79)(32 58 80)(37 87 45)(38 88 46)(39 85 47)(40 86 48)(49 68 72)(50 65 69)(51 66 70)(52 67 71)(89 139 143)(90 140 144)(91 137 141)(92 138 142)(93 98 124)(94 99 121)(95 100 122)(96 97 123)
(1 26 44)(2 27 41)(3 28 42)(4 25 43)(5 127 131)(6 128 132)(7 125 129)(8 126 130)(9 18 63)(10 19 64)(11 20 61)(12 17 62)(13 53 36)(14 54 33)(15 55 34)(16 56 35)(21 109 82)(22 110 83)(23 111 84)(24 112 81)(29 77 59)(30 78 60)(31 79 57)(32 80 58)(37 45 87)(38 46 88)(39 47 85)(40 48 86)(49 68 72)(50 65 69)(51 66 70)(52 67 71)(73 133 106)(74 134 107)(75 135 108)(76 136 105)(89 139 143)(90 140 144)(91 137 141)(92 138 142)(93 98 124)(94 99 121)(95 100 122)(96 97 123)(101 113 117)(102 114 118)(103 115 119)(104 116 120)
(1 111 67)(2 112 68)(3 109 65)(4 110 66)(5 97 103)(6 98 104)(7 99 101)(8 100 102)(9 60 15)(10 57 16)(11 58 13)(12 59 14)(17 29 54)(18 30 55)(19 31 56)(20 32 53)(21 50 42)(22 51 43)(23 52 44)(24 49 41)(25 83 70)(26 84 71)(27 81 72)(28 82 69)(33 62 77)(34 63 78)(35 64 79)(36 61 80)(37 139 135)(38 140 136)(39 137 133)(40 138 134)(45 143 108)(46 144 105)(47 141 106)(48 142 107)(73 85 91)(74 86 92)(75 87 89)(76 88 90)(93 120 132)(94 117 129)(95 118 130)(96 119 131)(113 125 121)(114 126 122)(115 127 123)(116 128 124)
G:=sub<Sym(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), (1,114,11,133)(2,113,12,136)(3,116,9,135)(4,115,10,134)(5,79,86,22)(6,78,87,21)(7,77,88,24)(8,80,85,23)(13,137,67,122)(14,140,68,121)(15,139,65,124)(16,138,66,123)(17,105,27,117)(18,108,28,120)(19,107,25,119)(20,106,26,118)(29,46,81,129)(30,45,82,132)(31,48,83,131)(32,47,84,130)(33,90,49,99)(34,89,50,98)(35,92,51,97)(36,91,52,100)(37,109,128,60)(38,112,125,59)(39,111,126,58)(40,110,127,57)(41,101,62,76)(42,104,63,75)(43,103,64,74)(44,102,61,73)(53,141,71,95)(54,144,72,94)(55,143,69,93)(56,142,70,96), (5,131,127)(6,132,128)(7,129,125)(8,130,126)(13,53,36)(14,54,33)(15,55,34)(16,56,35)(21,82,109)(22,83,110)(23,84,111)(24,81,112)(29,59,77)(30,60,78)(31,57,79)(32,58,80)(37,87,45)(38,88,46)(39,85,47)(40,86,48)(49,68,72)(50,65,69)(51,66,70)(52,67,71)(89,139,143)(90,140,144)(91,137,141)(92,138,142)(93,98,124)(94,99,121)(95,100,122)(96,97,123), (1,26,44)(2,27,41)(3,28,42)(4,25,43)(5,127,131)(6,128,132)(7,125,129)(8,126,130)(9,18,63)(10,19,64)(11,20,61)(12,17,62)(13,53,36)(14,54,33)(15,55,34)(16,56,35)(21,109,82)(22,110,83)(23,111,84)(24,112,81)(29,77,59)(30,78,60)(31,79,57)(32,80,58)(37,45,87)(38,46,88)(39,47,85)(40,48,86)(49,68,72)(50,65,69)(51,66,70)(52,67,71)(73,133,106)(74,134,107)(75,135,108)(76,136,105)(89,139,143)(90,140,144)(91,137,141)(92,138,142)(93,98,124)(94,99,121)(95,100,122)(96,97,123)(101,113,117)(102,114,118)(103,115,119)(104,116,120), (1,111,67)(2,112,68)(3,109,65)(4,110,66)(5,97,103)(6,98,104)(7,99,101)(8,100,102)(9,60,15)(10,57,16)(11,58,13)(12,59,14)(17,29,54)(18,30,55)(19,31,56)(20,32,53)(21,50,42)(22,51,43)(23,52,44)(24,49,41)(25,83,70)(26,84,71)(27,81,72)(28,82,69)(33,62,77)(34,63,78)(35,64,79)(36,61,80)(37,139,135)(38,140,136)(39,137,133)(40,138,134)(45,143,108)(46,144,105)(47,141,106)(48,142,107)(73,85,91)(74,86,92)(75,87,89)(76,88,90)(93,120,132)(94,117,129)(95,118,130)(96,119,131)(113,125,121)(114,126,122)(115,127,123)(116,128,124)>;
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)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,114,11,133)(2,113,12,136)(3,116,9,135)(4,115,10,134)(5,79,86,22)(6,78,87,21)(7,77,88,24)(8,80,85,23)(13,137,67,122)(14,140,68,121)(15,139,65,124)(16,138,66,123)(17,105,27,117)(18,108,28,120)(19,107,25,119)(20,106,26,118)(29,46,81,129)(30,45,82,132)(31,48,83,131)(32,47,84,130)(33,90,49,99)(34,89,50,98)(35,92,51,97)(36,91,52,100)(37,109,128,60)(38,112,125,59)(39,111,126,58)(40,110,127,57)(41,101,62,76)(42,104,63,75)(43,103,64,74)(44,102,61,73)(53,141,71,95)(54,144,72,94)(55,143,69,93)(56,142,70,96), (5,131,127)(6,132,128)(7,129,125)(8,130,126)(13,53,36)(14,54,33)(15,55,34)(16,56,35)(21,82,109)(22,83,110)(23,84,111)(24,81,112)(29,59,77)(30,60,78)(31,57,79)(32,58,80)(37,87,45)(38,88,46)(39,85,47)(40,86,48)(49,68,72)(50,65,69)(51,66,70)(52,67,71)(89,139,143)(90,140,144)(91,137,141)(92,138,142)(93,98,124)(94,99,121)(95,100,122)(96,97,123), (1,26,44)(2,27,41)(3,28,42)(4,25,43)(5,127,131)(6,128,132)(7,125,129)(8,126,130)(9,18,63)(10,19,64)(11,20,61)(12,17,62)(13,53,36)(14,54,33)(15,55,34)(16,56,35)(21,109,82)(22,110,83)(23,111,84)(24,112,81)(29,77,59)(30,78,60)(31,79,57)(32,80,58)(37,45,87)(38,46,88)(39,47,85)(40,48,86)(49,68,72)(50,65,69)(51,66,70)(52,67,71)(73,133,106)(74,134,107)(75,135,108)(76,136,105)(89,139,143)(90,140,144)(91,137,141)(92,138,142)(93,98,124)(94,99,121)(95,100,122)(96,97,123)(101,113,117)(102,114,118)(103,115,119)(104,116,120), (1,111,67)(2,112,68)(3,109,65)(4,110,66)(5,97,103)(6,98,104)(7,99,101)(8,100,102)(9,60,15)(10,57,16)(11,58,13)(12,59,14)(17,29,54)(18,30,55)(19,31,56)(20,32,53)(21,50,42)(22,51,43)(23,52,44)(24,49,41)(25,83,70)(26,84,71)(27,81,72)(28,82,69)(33,62,77)(34,63,78)(35,64,79)(36,61,80)(37,139,135)(38,140,136)(39,137,133)(40,138,134)(45,143,108)(46,144,105)(47,141,106)(48,142,107)(73,85,91)(74,86,92)(75,87,89)(76,88,90)(93,120,132)(94,117,129)(95,118,130)(96,119,131)(113,125,121)(114,126,122)(115,127,123)(116,128,124) );
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),(125,126,127,128),(129,130,131,132),(133,134,135,136),(137,138,139,140),(141,142,143,144)], [(1,114,11,133),(2,113,12,136),(3,116,9,135),(4,115,10,134),(5,79,86,22),(6,78,87,21),(7,77,88,24),(8,80,85,23),(13,137,67,122),(14,140,68,121),(15,139,65,124),(16,138,66,123),(17,105,27,117),(18,108,28,120),(19,107,25,119),(20,106,26,118),(29,46,81,129),(30,45,82,132),(31,48,83,131),(32,47,84,130),(33,90,49,99),(34,89,50,98),(35,92,51,97),(36,91,52,100),(37,109,128,60),(38,112,125,59),(39,111,126,58),(40,110,127,57),(41,101,62,76),(42,104,63,75),(43,103,64,74),(44,102,61,73),(53,141,71,95),(54,144,72,94),(55,143,69,93),(56,142,70,96)], [(5,131,127),(6,132,128),(7,129,125),(8,130,126),(13,53,36),(14,54,33),(15,55,34),(16,56,35),(21,82,109),(22,83,110),(23,84,111),(24,81,112),(29,59,77),(30,60,78),(31,57,79),(32,58,80),(37,87,45),(38,88,46),(39,85,47),(40,86,48),(49,68,72),(50,65,69),(51,66,70),(52,67,71),(89,139,143),(90,140,144),(91,137,141),(92,138,142),(93,98,124),(94,99,121),(95,100,122),(96,97,123)], [(1,26,44),(2,27,41),(3,28,42),(4,25,43),(5,127,131),(6,128,132),(7,125,129),(8,126,130),(9,18,63),(10,19,64),(11,20,61),(12,17,62),(13,53,36),(14,54,33),(15,55,34),(16,56,35),(21,109,82),(22,110,83),(23,111,84),(24,112,81),(29,77,59),(30,78,60),(31,79,57),(32,80,58),(37,45,87),(38,46,88),(39,47,85),(40,48,86),(49,68,72),(50,65,69),(51,66,70),(52,67,71),(73,133,106),(74,134,107),(75,135,108),(76,136,105),(89,139,143),(90,140,144),(91,137,141),(92,138,142),(93,98,124),(94,99,121),(95,100,122),(96,97,123),(101,113,117),(102,114,118),(103,115,119),(104,116,120)], [(1,111,67),(2,112,68),(3,109,65),(4,110,66),(5,97,103),(6,98,104),(7,99,101),(8,100,102),(9,60,15),(10,57,16),(11,58,13),(12,59,14),(17,29,54),(18,30,55),(19,31,56),(20,32,53),(21,50,42),(22,51,43),(23,52,44),(24,49,41),(25,83,70),(26,84,71),(27,81,72),(28,82,69),(33,62,77),(34,63,78),(35,64,79),(36,61,80),(37,139,135),(38,140,136),(39,137,133),(40,138,134),(45,143,108),(46,144,105),(47,141,106),(48,142,107),(73,85,91),(74,86,92),(75,87,89),(76,88,90),(93,120,132),(94,117,129),(95,118,130),(96,119,131),(113,125,121),(114,126,122),(115,127,123),(116,128,124)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3J | 4A | ··· | 4F | 6A | ··· | 6F | 6G | ··· | 6AD | 12A | ··· | 12L | 12M | ··· | 12BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | He3 | C2×He3 | C4×He3 | D4×He3 | Q8×He3 |
| kernel | C4⋊C4×He3 | C2×C4×He3 | C32×C4⋊C4 | C4×He3 | C6×C12 | C3×C12 | C2×He3 | C2×He3 | C3×C6 | C3×C6 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 3 | 8 | 4 | 24 | 32 | 1 | 1 | 8 | 8 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of C4⋊C4×He3 ►in GL5(𝔽13)
| 0 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 5 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [0,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,5,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C4⋊C4×He3 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times {\rm He}_3 % in TeX
G:=Group("C4:C4xHe3"); // GroupNames label
G:=SmallGroup(432,207);
// by ID
G=gap.SmallGroup(432,207);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,260,1109]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^3=d^3=e^3=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations