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## G = C22×C42.C2order 128 = 27

### Direct product of C22 and C42.C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C22×C42.C2
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=cd2, ede-1=c2d >

Subgroups: 716 in 596 conjugacy classes, 476 normal (7 characteristic)
C1, C2, C2 [×14], C4 [×8], C4 [×24], C22, C22 [×34], C2×C4 [×52], C2×C4 [×72], C23 [×15], C42 [×16], C4⋊C4 [×96], C22×C4 [×50], C22×C4 [×24], C24, C2×C42 [×12], C2×C4⋊C4 [×72], C42.C2 [×64], C23×C4, C23×C4 [×6], C22×C42, C22×C4⋊C4 [×6], C2×C42.C2 [×24], C22×C42.C2
Quotients: C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C4○D4 [×8], C24 [×31], C42.C2 [×16], C22×Q8 [×14], C2×C4○D4 [×12], C25, C2×C42.C2 [×12], Q8×C23, C22×C4○D4 [×2], C22×C42.C2

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

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

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

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

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 2 2 type + + + + - image C1 C2 C2 C2 Q8 C4○D4 kernel C22×C42.C2 C22×C42 C22×C4⋊C4 C2×C42.C2 C22×C4 C23 # reps 1 1 6 24 8 16

Matrix representation of C22×C42.C2 in GL7(𝔽5)

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

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,4,1,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C22×C42.C2 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,2169);
// by ID

G=gap.SmallGroup(128,2169);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,184]);
// 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*d^2,e*d*e^-1=c^2*d>;
// generators/relations

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