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## G = C2×C42⋊9C4order 128 = 27

### Direct product of C2 and C42⋊9C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C42⋊9C4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C2×C42⋊9C4
 Lower central C1 — C22 — C2×C42⋊9C4
 Upper central C1 — C24 — C2×C42⋊9C4
 Jennings C1 — C23 — C2×C42⋊9C4

Generators and relations for C2×C429C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 636 in 444 conjugacy classes, 300 normal (7 characteristic)
C1, C2, C2 [×14], C4 [×24], C4 [×8], C22, C22 [×34], C2×C4 [×84], C2×C4 [×40], C23, C23 [×14], C42 [×16], C4⋊C4 [×48], C22×C4 [×50], C22×C4 [×24], C24, C2×C42 [×12], C2×C4⋊C4 [×24], C2×C4⋊C4 [×24], C23×C4 [×7], C429C4 [×8], C22×C42, C22×C4⋊C4 [×6], C2×C429C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×12], Q8 [×12], C23 [×15], C4⋊C4 [×48], C22×C4 [×14], C2×D4 [×18], C2×Q8 [×18], C24, C2×C4⋊C4 [×36], C41D4 [×4], C4⋊Q8 [×12], C23×C4, C22×D4 [×3], C22×Q8 [×3], C429C4 [×8], C22×C4⋊C4 [×3], C2×C41D4, C2×C4⋊Q8 [×3], C2×C429C4

Smallest permutation representation of C2×C429C4
Regular action on 128 points
Generators in S128
(1 28)(2 25)(3 26)(4 27)(5 92)(6 89)(7 90)(8 91)(9 20)(10 17)(11 18)(12 19)(13 77)(14 78)(15 79)(16 80)(21 40)(22 37)(23 38)(24 39)(29 99)(30 100)(31 97)(32 98)(33 53)(34 54)(35 55)(36 56)(41 112)(42 109)(43 110)(44 111)(45 116)(46 113)(47 114)(48 115)(49 73)(50 74)(51 75)(52 76)(57 61)(58 62)(59 63)(60 64)(65 121)(66 122)(67 123)(68 124)(69 117)(70 118)(71 119)(72 120)(81 107)(82 108)(83 105)(84 106)(85 103)(86 104)(87 101)(88 102)(93 125)(94 126)(95 127)(96 128)
(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)
(1 29 22 52)(2 30 23 49)(3 31 24 50)(4 32 21 51)(5 66 48 105)(6 67 45 106)(7 68 46 107)(8 65 47 108)(9 78 35 61)(10 79 36 62)(11 80 33 63)(12 77 34 64)(13 54 60 19)(14 55 57 20)(15 56 58 17)(16 53 59 18)(25 100 38 73)(26 97 39 74)(27 98 40 75)(28 99 37 76)(41 102 125 71)(42 103 126 72)(43 104 127 69)(44 101 128 70)(81 90 124 113)(82 91 121 114)(83 92 122 115)(84 89 123 116)(85 94 120 109)(86 95 117 110)(87 96 118 111)(88 93 119 112)
(1 108 9 119)(2 107 10 118)(3 106 11 117)(4 105 12 120)(5 64 109 51)(6 63 110 50)(7 62 111 49)(8 61 112 52)(13 126 98 115)(14 125 99 114)(15 128 100 113)(16 127 97 116)(17 70 25 81)(18 69 26 84)(19 72 27 83)(20 71 28 82)(21 66 34 85)(22 65 35 88)(23 68 36 87)(24 67 33 86)(29 47 78 93)(30 46 79 96)(31 45 80 95)(32 48 77 94)(37 121 55 102)(38 124 56 101)(39 123 53 104)(40 122 54 103)(41 76 91 57)(42 75 92 60)(43 74 89 59)(44 73 90 58)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4X 4Y ··· 4AN order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 2 type + + + + + - image C1 C2 C2 C2 C4 D4 Q8 kernel C2×C42⋊9C4 C42⋊9C4 C22×C42 C22×C4⋊C4 C2×C42 C22×C4 C22×C4 # reps 1 8 1 6 16 12 12

Matrix representation of C2×C429C4 in GL6(𝔽5)

 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 2
,
 3 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;

C2×C429C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_9C_4
% in TeX

G:=Group("C2xC4^2:9C4");
// GroupNames label

G:=SmallGroup(128,1016);
// by ID

G=gap.SmallGroup(128,1016);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,184]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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