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## G = C4⋊C4×He3order 432 = 24·33

### Direct product of C4⋊C4 and He3

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4⋊C4×He3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C22×He3 — C2×C4×He3 — C4⋊C4×He3
 Lower central C1 — C6 — C4⋊C4×He3
 Upper central C1 — C2×C6 — C4⋊C4×He3

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

Smallest permutation representation of 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

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