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## G = C22×C2.C42order 128 = 27

### Direct product of C22 and C2.C42

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C22×C2.C42
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C22×C2.C42
 Lower central C1 — C2 — C22×C2.C42
 Upper central C1 — C25 — C22×C2.C42
 Jennings C1 — C23 — C22×C2.C42

Generators and relations for C22×C2.C42
G = < a,b,c,d,e | a2=b2=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec >

Subgroups: 1324 in 940 conjugacy classes, 556 normal (6 characteristic)
C1, C2, C2 [×30], C4 [×24], C22, C22 [×154], C2×C4 [×24], C2×C4 [×168], C23, C23 [×154], C22×C4 [×84], C22×C4 [×168], C24 [×31], C2.C42 [×16], C23×C4 [×42], C23×C4 [×24], C25, C2×C2.C42 [×12], C24×C4 [×3], C22×C2.C42
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], D4 [×12], Q8 [×4], C23 [×15], C42 [×16], C22⋊C4 [×48], C4⋊C4 [×48], C22×C4 [×42], C2×D4 [×18], C2×Q8 [×6], C24, C2.C42 [×64], C2×C42 [×12], C2×C22⋊C4 [×36], C2×C4⋊C4 [×36], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C2.C42 [×24], C22×C42, C22×C22⋊C4 [×3], C22×C4⋊C4 [×3], C22×C2.C42

Smallest permutation representation of C22×C2.C42
Regular action on 128 points
Generators in S128
(1 61)(2 62)(3 63)(4 64)(5 39)(6 40)(7 37)(8 38)(9 47)(10 48)(11 45)(12 46)(13 35)(14 36)(15 33)(16 34)(17 55)(18 56)(19 53)(20 54)(21 43)(22 44)(23 41)(24 42)(25 115)(26 116)(27 113)(28 114)(29 51)(30 52)(31 49)(32 50)(57 90)(58 91)(59 92)(60 89)(65 104)(66 101)(67 102)(68 103)(69 108)(70 105)(71 106)(72 107)(73 112)(74 109)(75 110)(76 111)(77 119)(78 120)(79 117)(80 118)(81 123)(82 124)(83 121)(84 122)(85 127)(86 128)(87 125)(88 126)(93 99)(94 100)(95 97)(96 98)
(1 31)(2 32)(3 29)(4 30)(5 57)(6 58)(7 59)(8 60)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 77)(26 78)(27 79)(28 80)(33 45)(34 46)(35 47)(36 48)(37 92)(38 89)(39 90)(40 91)(41 53)(42 54)(43 55)(44 56)(49 61)(50 62)(51 63)(52 64)(65 94)(66 95)(67 96)(68 93)(69 73)(70 74)(71 75)(72 76)(81 85)(82 86)(83 87)(84 88)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)
(1 95)(2 96)(3 93)(4 94)(5 81)(6 82)(7 83)(8 84)(9 25)(10 26)(11 27)(12 28)(13 77)(14 78)(15 79)(16 80)(17 72)(18 69)(19 70)(20 71)(21 76)(22 73)(23 74)(24 75)(29 68)(30 65)(31 66)(32 67)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 85)(58 86)(59 87)(60 88)(61 97)(62 98)(63 99)(64 100)(89 126)(90 127)(91 128)(92 125)
(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)
(1 119 111 7)(2 36 112 84)(3 117 109 5)(4 34 110 82)(6 94 118 42)(8 96 120 44)(9 17 125 101)(10 69 126 50)(11 19 127 103)(12 71 128 52)(13 21 121 97)(14 73 122 62)(15 23 123 99)(16 75 124 64)(18 89 102 26)(20 91 104 28)(22 38 98 78)(24 40 100 80)(25 72 92 49)(27 70 90 51)(29 113 105 57)(30 46 106 86)(31 115 107 59)(32 48 108 88)(33 41 81 93)(35 43 83 95)(37 61 77 76)(39 63 79 74)(45 53 85 68)(47 55 87 66)(54 58 65 114)(56 60 67 116)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2AE 4A ··· 4AV order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 2 2 type + + + + - image C1 C2 C2 C4 D4 Q8 kernel C22×C2.C42 C2×C2.C42 C24×C4 C23×C4 C24 C24 # reps 1 12 3 48 12 4

Matrix representation of C22×C2.C42 in GL6(𝔽5)

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

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

C22×C2.C42 in GAP, Magma, Sage, TeX

C_2^2\times C_2.C_4^2
% in TeX

G:=Group("C2^2xC2.C4^2");
// GroupNames label

G:=SmallGroup(128,998);
// by ID

G=gap.SmallGroup(128,998);
# by ID

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

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

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