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## G = C22×C8⋊C4order 128 = 27

### Direct product of C22 and C8⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C22×C8⋊C4
G = < a,b,c,d | a2=b2=c8=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 380 in 340 conjugacy classes, 300 normal (8 characteristic)
C1, C2, C2 [×14], C4 [×8], C4 [×8], C22, C22 [×34], C8 [×16], C2×C4, C2×C4 [×35], C2×C4 [×24], C23 [×15], C42 [×16], C2×C8 [×56], C22×C4 [×26], C22×C4 [×8], C24, C8⋊C4 [×16], C2×C42 [×12], C22×C8 [×28], C23×C4, C23×C4 [×2], C2×C8⋊C4 [×12], C22×C42, C23×C8 [×2], C22×C8⋊C4
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], M4(2) [×8], C22×C4 [×42], C24, C8⋊C4 [×16], C2×C42 [×12], C2×M4(2) [×12], C23×C4 [×3], C2×C8⋊C4 [×12], C22×C42, C22×M4(2) [×2], C22×C8⋊C4

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

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

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

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

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 type + + + + image C1 C2 C2 C2 C4 C4 C4 M4(2) kernel C22×C8⋊C4 C2×C8⋊C4 C22×C42 C23×C8 C2×C42 C22×C8 C23×C4 C23 # reps 1 12 1 2 12 32 4 16

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

 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 13 0 0 0 0 0 1 0 0 0 0 0 13 0 0 0 0 0 2 5 0 0 0 12 15
,
 4 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,2,12,0,0,0,5,15],[4,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C22×C8⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC8:C4");
// GroupNames label

G:=SmallGroup(128,1602);
// by ID

G=gap.SmallGroup(128,1602);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,925,232,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=d^4=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^5>;
// generators/relations

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