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## G = C22×C4⋊Q8order 128 = 27

### Direct product of C22 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22×C4⋊Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C22×C4⋊Q8
 Lower central C1 — C22 — C22×C4⋊Q8
 Upper central C1 — C24 — C22×C4⋊Q8
 Jennings C1 — C22 — C22×C4⋊Q8

Generators and relations for C22×C4⋊Q8
G = < a,b,c,d,e | a2=b2=c4=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 1068 in 812 conjugacy classes, 556 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C42, C2×C4⋊C4, C4⋊Q8, C23×C4, C23×C4, C22×Q8, C22×Q8, C22×C42, C22×C4⋊C4, C2×C4⋊Q8, Q8×C23, C22×C4⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C22×D4, C22×Q8, C25, C2×C4⋊Q8, D4×C23, Q8×C23, C22×C4⋊Q8

Smallest permutation representation of C22×C4⋊Q8
Regular action on 128 points
Generators in S128
(1 24)(2 21)(3 22)(4 23)(5 56)(6 53)(7 54)(8 55)(9 48)(10 45)(11 46)(12 47)(13 64)(14 61)(15 62)(16 63)(17 35)(18 36)(19 33)(20 34)(25 38)(26 39)(27 40)(28 37)(29 43)(30 44)(31 41)(32 42)(49 124)(50 121)(51 122)(52 123)(57 84)(58 81)(59 82)(60 83)(65 107)(66 108)(67 105)(68 106)(69 117)(70 118)(71 119)(72 120)(73 113)(74 114)(75 115)(76 116)(77 125)(78 126)(79 127)(80 128)(85 102)(86 103)(87 104)(88 101)(89 98)(90 99)(91 100)(92 97)(93 109)(94 110)(95 111)(96 112)
(1 18)(2 19)(3 20)(4 17)(5 46)(6 47)(7 48)(8 45)(9 54)(10 55)(11 56)(12 53)(13 58)(14 59)(15 60)(16 57)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(29 40)(30 37)(31 38)(32 39)(49 126)(50 127)(51 128)(52 125)(61 82)(62 83)(63 84)(64 81)(65 111)(66 112)(67 109)(68 110)(69 115)(70 116)(71 113)(72 114)(73 119)(74 120)(75 117)(76 118)(77 123)(78 124)(79 121)(80 122)(85 100)(86 97)(87 98)(88 99)(89 104)(90 101)(91 102)(92 103)(93 105)(94 106)(95 107)(96 108)
(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 32 57 54)(2 29 58 55)(3 30 59 56)(4 31 60 53)(5 22 44 82)(6 23 41 83)(7 24 42 84)(8 21 43 81)(9 18 39 16)(10 19 40 13)(11 20 37 14)(12 17 38 15)(25 62 47 35)(26 63 48 36)(27 64 45 33)(28 61 46 34)(49 67 90 71)(50 68 91 72)(51 65 92 69)(52 66 89 70)(73 78 93 88)(74 79 94 85)(75 80 95 86)(76 77 96 87)(97 117 122 107)(98 118 123 108)(99 119 124 105)(100 120 121 106)(101 113 126 109)(102 114 127 110)(103 115 128 111)(104 116 125 112)
(1 127 57 102)(2 126 58 101)(3 125 59 104)(4 128 60 103)(5 96 44 76)(6 95 41 75)(7 94 42 74)(8 93 43 73)(9 68 39 72)(10 67 40 71)(11 66 37 70)(12 65 38 69)(13 90 19 49)(14 89 20 52)(15 92 17 51)(16 91 18 50)(21 78 81 88)(22 77 82 87)(23 80 83 86)(24 79 84 85)(25 117 47 107)(26 120 48 106)(27 119 45 105)(28 118 46 108)(29 113 55 109)(30 116 56 112)(31 115 53 111)(32 114 54 110)(33 124 64 99)(34 123 61 98)(35 122 62 97)(36 121 63 100)

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

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

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

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 C2 D4 Q8 kernel C22×C4⋊Q8 C22×C42 C22×C4⋊C4 C2×C4⋊Q8 Q8×C23 C22×C4 C22×C4 # reps 1 1 4 24 2 8 16

Matrix representation of C22×C4⋊Q8 in GL6(𝔽5)

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

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

C22×C4⋊Q8 in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes Q_8
% in TeX

G:=Group("C2^2xC4:Q8");
// GroupNames label

G:=SmallGroup(128,2173);
// by ID

G=gap.SmallGroup(128,2173);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,352]);
// Polycyclic

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

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