Copied to
clipboard

## G = C22×C4⋊C8order 128 = 27

### Direct product of C22 and C4⋊C8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C22×C4⋊C8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C22×C4⋊C8
 Lower central C1 — C2 — C22×C4⋊C8
 Upper central C1 — C23×C4 — C22×C4⋊C8
 Jennings C1 — C2 — C2 — C2×C4 — C22×C4⋊C8

Generators and relations for C22×C4⋊C8
G = < a,b,c,d | a2=b2=c4=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 380 in 320 conjugacy classes, 260 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×14], C4 [×4], C22, C22 [×34], C8 [×8], C2×C4, C2×C4 [×59], C2×C4 [×12], C23 [×15], C42 [×16], C2×C8 [×8], C2×C8 [×24], C22×C4 [×34], C22×C4 [×4], C24, C4⋊C8 [×16], C2×C42 [×12], C22×C8 [×12], C22×C8 [×8], C23×C4 [×3], C2×C4⋊C8 [×12], C22×C42, C23×C8 [×2], C22×C4⋊C8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C2×C8 [×28], M4(2) [×4], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C4⋊C8 [×16], C2×C4⋊C4 [×12], C22×C8 [×14], C2×M4(2) [×6], C23×C4, C22×D4, C22×Q8, C2×C4⋊C8 [×12], C22×C4⋊C4, C23×C8, C22×M4(2), C22×C4⋊C8

Smallest permutation representation of C22×C4⋊C8
Regular action on 128 points
Generators in S128
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 121)(16 122)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 108)(26 109)(27 110)(28 111)(29 112)(30 105)(31 106)(32 107)(41 71)(42 72)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(73 87)(74 88)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(89 119)(90 120)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(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)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 93)(66 94)(67 95)(68 96)(69 89)(70 90)(71 91)(72 92)(81 125)(82 126)(83 127)(84 128)(85 121)(86 122)(87 123)(88 124)
(1 115 111 77)(2 78 112 116)(3 117 105 79)(4 80 106 118)(5 119 107 73)(6 74 108 120)(7 113 109 75)(8 76 110 114)(9 53 47 35)(10 36 48 54)(11 55 41 37)(12 38 42 56)(13 49 43 39)(14 40 44 50)(15 51 45 33)(16 34 46 52)(17 66 100 128)(18 121 101 67)(19 68 102 122)(20 123 103 69)(21 70 104 124)(22 125 97 71)(23 72 98 126)(24 127 99 65)(25 90 64 88)(26 81 57 91)(27 92 58 82)(28 83 59 93)(29 94 60 84)(30 85 61 95)(31 96 62 86)(32 87 63 89)
(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)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4P 4Q ··· 4AF 8A ··· 8AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C8 D4 Q8 M4(2) kernel C22×C4⋊C8 C2×C4⋊C8 C22×C42 C23×C8 C2×C42 C23×C4 C22×C4 C22×C4 C22×C4 C23 # reps 1 12 1 2 12 4 32 4 4 8

Matrix representation of C22×C4⋊C8 in GL5(𝔽17)

 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 15 0 0 0 1 1
,
 2 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 16 16

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,1,0,0,0,15,1],[2,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,0,16] >;

C22×C4⋊C8 in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes C_8
% in TeX

G:=Group("C2^2xC4:C8");
// GroupNames label

G:=SmallGroup(128,1634);
// by ID

G=gap.SmallGroup(128,1634);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^4=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽