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## G = C22×Q8⋊C4order 128 = 27

### Direct product of C22 and Q8⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C22×Q8⋊C4
G = < a,b,c,d,e | a2=b2=c4=e4=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c-1d >

Subgroups: 636 in 408 conjugacy classes, 220 normal (14 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Q8⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×C8, C23×C4, C23×C4, C22×Q8, C22×Q8, C2×Q8⋊C4, C22×C4⋊C4, C23×C8, Q8×C23, C22×Q8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C24, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C23×C4, C22×D4, C2×Q8⋊C4, C22×C22⋊C4, C22×SD16, C22×Q16, C22×Q8⋊C4

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

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

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

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

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 1 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C4 D4 D4 SD16 Q16 kernel C22×Q8⋊C4 C2×Q8⋊C4 C22×C4⋊C4 C23×C8 Q8×C23 C22×Q8 C22×C4 C24 C23 C23 # reps 1 12 1 1 1 16 7 1 8 8

Matrix representation of C22×Q8⋊C4 in GL7(𝔽17)

 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 16 0
,
 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 16 0 0 0 0 0 0 0 7 1 0 0 0 0 0 1 10
,
 16 0 0 0 0 0 0 0 4 7 0 0 0 0 0 10 13 0 0 0 0 0 0 0 12 5 0 0 0 0 0 5 5 0 0 0 0 0 0 0 6 4 0 0 0 0 0 4 11

G:=sub<GL(7,GF(17))| [16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,7,1,0,0,0,0,0,1,10],[16,0,0,0,0,0,0,0,4,10,0,0,0,0,0,7,13,0,0,0,0,0,0,0,12,5,0,0,0,0,0,5,5,0,0,0,0,0,0,0,6,4,0,0,0,0,0,4,11] >;

C22×Q8⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xQ8:C4");
// GroupNames label

G:=SmallGroup(128,1623);
// by ID

G=gap.SmallGroup(128,1623);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,2804,1411,172]);
// Polycyclic

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

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