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## G = C22×C2.D8order 128 = 27

### Direct product of C22 and C2.D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C22×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C22×C2.D8
 Lower central C1 — C2 — C4 — C22×C2.D8
 Upper central C1 — C24 — C23×C4 — C22×C2.D8
 Jennings C1 — C2 — C2 — C2×C4 — C22×C2.D8

Generators and relations for C22×C2.D8
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 460 in 300 conjugacy classes, 220 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C23×C4, C2×C2.D8, C22×C4⋊C4, C23×C8, C22×C2.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C24, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C23×C4, C22×D4, C22×Q8, C2×C2.D8, C22×C4⋊C4, C22×D8, C22×Q16, C22×C2.D8

Smallest permutation representation of C22×C2.D8
Regular action on 128 points
Generators in S128
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 81)(32 82)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(49 70)(50 71)(51 72)(52 65)(53 66)(54 67)(55 68)(56 69)(73 95)(74 96)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(97 116)(98 117)(99 118)(100 119)(101 120)(102 113)(103 114)(104 115)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 121)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 124)(50 125)(51 126)(52 127)(53 128)(54 121)(55 122)(56 123)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(73 117)(74 118)(75 119)(76 120)(77 113)(78 114)(79 115)(80 116)(89 100)(90 101)(91 102)(92 103)(93 104)(94 97)(95 98)(96 99)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 81)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 112)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 121)(96 122)
(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)
(1 116 59 66)(2 115 60 65)(3 114 61 72)(4 113 62 71)(5 120 63 70)(6 119 64 69)(7 118 57 68)(8 117 58 67)(9 103 38 51)(10 102 39 50)(11 101 40 49)(12 100 33 56)(13 99 34 55)(14 98 35 54)(15 97 36 53)(16 104 37 52)(17 77 82 108)(18 76 83 107)(19 75 84 106)(20 74 85 105)(21 73 86 112)(22 80 87 111)(23 79 88 110)(24 78 81 109)(25 124 43 90)(26 123 44 89)(27 122 45 96)(28 121 46 95)(29 128 47 94)(30 127 48 93)(31 126 41 92)(32 125 42 91)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4H 4I ··· 4X 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C4 D4 Q8 D4 D8 Q16 kernel C22×C2.D8 C2×C2.D8 C22×C4⋊C4 C23×C8 C22×C8 C22×C4 C22×C4 C24 C23 C23 # reps 1 12 2 1 16 3 4 1 8 8

Matrix representation of C22×C2.D8 in GL5(𝔽17)

 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
,
 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 8 0 0 0 0 0 15
,
 16 0 0 0 0 0 1 0 0 0 0 0 13 0 0 0 0 0 0 15 0 0 0 9 0

G:=sub<GL(5,GF(17))| [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],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,8,0,0,0,0,0,15],[16,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,0,9,0,0,0,15,0] >;

C22×C2.D8 in GAP, Magma, Sage, TeX

C_2^2\times C_2.D_8
% in TeX

G:=Group("C2^2xC2.D8");
// GroupNames label

G:=SmallGroup(128,1640);
// by ID

G=gap.SmallGroup(128,1640);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,568,2804,172]);
// Polycyclic

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

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