direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23×C4⋊C4, C24.24Q8, C24.194D4, C22.6C25, C25.99C22, C24.653C23, C23.266C24, C4⋊2(C23×C4), (C23×C4)⋊18C4, (C24×C4).8C2, C2.2(C24×C4), C2.2(D4×C23), C2.1(Q8×C23), C24.137(C2×C4), (C2×C4).153C24, C23.887(C2×D4), C23.147(C2×Q8), C22.46(C23×C4), C22.48(C22×Q8), C23.295(C22×C4), (C23×C4).660C22, C22.155(C22×D4), (C22×C4).1290C23, (C22×C4)⋊52(C2×C4), (C2×C4)⋊12(C22×C4), SmallGroup(128,2152)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 1500 in 1244 conjugacy classes, 988 normal (8 characteristic)
C1, C2 [×3], C2 [×28], C4 [×16], C4 [×16], C22, C22 [×154], C2×C4 [×136], C2×C4 [×112], C23 [×155], C4⋊C4 [×64], C22×C4 [×196], C22×C4 [×112], C24 [×31], C2×C4⋊C4 [×112], C23×C4 [×58], C23×C4 [×16], C25, C22×C4⋊C4 [×28], C24×C4, C24×C4 [×2], C23×C4⋊C4
Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], D4 [×8], Q8 [×8], C23 [×155], C4⋊C4 [×64], C22×C4 [×140], C2×D4 [×28], C2×Q8 [×28], C24 [×31], C2×C4⋊C4 [×112], C23×C4 [×30], C22×D4 [×14], C22×Q8 [×14], C25, C22×C4⋊C4 [×28], C24×C4, D4×C23, Q8×C23, C23×C4⋊C4
Generators and relations
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, cd=dc, ce=ec, ede-1=d-1 >
►Regular action on 128 points
Generators in S128
(1 31)(2 32)(3 29)(4 30)(5 23)(6 24)(7 21)(8 22)(9 19)(10 20)(11 17)(12 18)(13 43)(14 44)(15 41)(16 42)(25 61)(26 62)(27 63)(28 64)(33 55)(34 56)(35 53)(36 54)(37 81)(38 82)(39 83)(40 84)(45 59)(46 60)(47 57)(48 58)(49 108)(50 105)(51 106)(52 107)(65 121)(66 122)(67 123)(68 124)(69 103)(70 104)(71 101)(72 102)(73 99)(74 100)(75 97)(76 98)(77 111)(78 112)(79 109)(80 110)(85 117)(86 118)(87 119)(88 120)(89 113)(90 114)(91 115)(92 116)(93 125)(94 126)(95 127)(96 128)
(1 63)(2 64)(3 61)(4 62)(5 9)(6 10)(7 11)(8 12)(13 83)(14 84)(15 81)(16 82)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 57)(34 58)(35 59)(36 60)(37 41)(38 42)(39 43)(40 44)(45 53)(46 54)(47 55)(48 56)(49 78)(50 79)(51 80)(52 77)(65 93)(66 94)(67 95)(68 96)(69 73)(70 74)(71 75)(72 76)(85 89)(86 90)(87 91)(88 92)(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 14)(2 15)(3 16)(4 13)(5 46)(6 47)(7 48)(8 45)(9 54)(10 55)(11 56)(12 53)(17 34)(18 35)(19 36)(20 33)(21 58)(22 59)(23 60)(24 57)(25 38)(26 39)(27 40)(28 37)(29 42)(30 43)(31 44)(32 41)(49 125)(50 126)(51 127)(52 128)(61 82)(62 83)(63 84)(64 81)(65 112)(66 109)(67 110)(68 111)(69 116)(70 113)(71 114)(72 115)(73 120)(74 117)(75 118)(76 119)(77 124)(78 121)(79 122)(80 123)(85 100)(86 97)(87 98)(88 99)(89 104)(90 101)(91 102)(92 103)(93 108)(94 105)(95 106)(96 107)
(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)
(1 106 18 113)(2 105 19 116)(3 108 20 115)(4 107 17 114)(5 88 28 79)(6 87 25 78)(7 86 26 77)(8 85 27 80)(9 92 32 50)(10 91 29 49)(11 90 30 52)(12 89 31 51)(13 96 34 71)(14 95 35 70)(15 94 36 69)(16 93 33 72)(21 118 62 111)(22 117 63 110)(23 120 64 109)(24 119 61 112)(37 122 46 99)(38 121 47 98)(39 124 48 97)(40 123 45 100)(41 126 54 103)(42 125 55 102)(43 128 56 101)(44 127 53 104)(57 76 82 65)(58 75 83 68)(59 74 84 67)(60 73 81 66)
G:=sub<Sym(128)| (1,31)(2,32)(3,29)(4,30)(5,23)(6,24)(7,21)(8,22)(9,19)(10,20)(11,17)(12,18)(13,43)(14,44)(15,41)(16,42)(25,61)(26,62)(27,63)(28,64)(33,55)(34,56)(35,53)(36,54)(37,81)(38,82)(39,83)(40,84)(45,59)(46,60)(47,57)(48,58)(49,108)(50,105)(51,106)(52,107)(65,121)(66,122)(67,123)(68,124)(69,103)(70,104)(71,101)(72,102)(73,99)(74,100)(75,97)(76,98)(77,111)(78,112)(79,109)(80,110)(85,117)(86,118)(87,119)(88,120)(89,113)(90,114)(91,115)(92,116)(93,125)(94,126)(95,127)(96,128), (1,63)(2,64)(3,61)(4,62)(5,9)(6,10)(7,11)(8,12)(13,83)(14,84)(15,81)(16,82)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,57)(34,58)(35,59)(36,60)(37,41)(38,42)(39,43)(40,44)(45,53)(46,54)(47,55)(48,56)(49,78)(50,79)(51,80)(52,77)(65,93)(66,94)(67,95)(68,96)(69,73)(70,74)(71,75)(72,76)(85,89)(86,90)(87,91)(88,92)(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), (1,14)(2,15)(3,16)(4,13)(5,46)(6,47)(7,48)(8,45)(9,54)(10,55)(11,56)(12,53)(17,34)(18,35)(19,36)(20,33)(21,58)(22,59)(23,60)(24,57)(25,38)(26,39)(27,40)(28,37)(29,42)(30,43)(31,44)(32,41)(49,125)(50,126)(51,127)(52,128)(61,82)(62,83)(63,84)(64,81)(65,112)(66,109)(67,110)(68,111)(69,116)(70,113)(71,114)(72,115)(73,120)(74,117)(75,118)(76,119)(77,124)(78,121)(79,122)(80,123)(85,100)(86,97)(87,98)(88,99)(89,104)(90,101)(91,102)(92,103)(93,108)(94,105)(95,106)(96,107), (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), (1,106,18,113)(2,105,19,116)(3,108,20,115)(4,107,17,114)(5,88,28,79)(6,87,25,78)(7,86,26,77)(8,85,27,80)(9,92,32,50)(10,91,29,49)(11,90,30,52)(12,89,31,51)(13,96,34,71)(14,95,35,70)(15,94,36,69)(16,93,33,72)(21,118,62,111)(22,117,63,110)(23,120,64,109)(24,119,61,112)(37,122,46,99)(38,121,47,98)(39,124,48,97)(40,123,45,100)(41,126,54,103)(42,125,55,102)(43,128,56,101)(44,127,53,104)(57,76,82,65)(58,75,83,68)(59,74,84,67)(60,73,81,66)>;
G:=Group( (1,31)(2,32)(3,29)(4,30)(5,23)(6,24)(7,21)(8,22)(9,19)(10,20)(11,17)(12,18)(13,43)(14,44)(15,41)(16,42)(25,61)(26,62)(27,63)(28,64)(33,55)(34,56)(35,53)(36,54)(37,81)(38,82)(39,83)(40,84)(45,59)(46,60)(47,57)(48,58)(49,108)(50,105)(51,106)(52,107)(65,121)(66,122)(67,123)(68,124)(69,103)(70,104)(71,101)(72,102)(73,99)(74,100)(75,97)(76,98)(77,111)(78,112)(79,109)(80,110)(85,117)(86,118)(87,119)(88,120)(89,113)(90,114)(91,115)(92,116)(93,125)(94,126)(95,127)(96,128), (1,63)(2,64)(3,61)(4,62)(5,9)(6,10)(7,11)(8,12)(13,83)(14,84)(15,81)(16,82)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,57)(34,58)(35,59)(36,60)(37,41)(38,42)(39,43)(40,44)(45,53)(46,54)(47,55)(48,56)(49,78)(50,79)(51,80)(52,77)(65,93)(66,94)(67,95)(68,96)(69,73)(70,74)(71,75)(72,76)(85,89)(86,90)(87,91)(88,92)(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), (1,14)(2,15)(3,16)(4,13)(5,46)(6,47)(7,48)(8,45)(9,54)(10,55)(11,56)(12,53)(17,34)(18,35)(19,36)(20,33)(21,58)(22,59)(23,60)(24,57)(25,38)(26,39)(27,40)(28,37)(29,42)(30,43)(31,44)(32,41)(49,125)(50,126)(51,127)(52,128)(61,82)(62,83)(63,84)(64,81)(65,112)(66,109)(67,110)(68,111)(69,116)(70,113)(71,114)(72,115)(73,120)(74,117)(75,118)(76,119)(77,124)(78,121)(79,122)(80,123)(85,100)(86,97)(87,98)(88,99)(89,104)(90,101)(91,102)(92,103)(93,108)(94,105)(95,106)(96,107), (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), (1,106,18,113)(2,105,19,116)(3,108,20,115)(4,107,17,114)(5,88,28,79)(6,87,25,78)(7,86,26,77)(8,85,27,80)(9,92,32,50)(10,91,29,49)(11,90,30,52)(12,89,31,51)(13,96,34,71)(14,95,35,70)(15,94,36,69)(16,93,33,72)(21,118,62,111)(22,117,63,110)(23,120,64,109)(24,119,61,112)(37,122,46,99)(38,121,47,98)(39,124,48,97)(40,123,45,100)(41,126,54,103)(42,125,55,102)(43,128,56,101)(44,127,53,104)(57,76,82,65)(58,75,83,68)(59,74,84,67)(60,73,81,66) );
G=PermutationGroup([(1,31),(2,32),(3,29),(4,30),(5,23),(6,24),(7,21),(8,22),(9,19),(10,20),(11,17),(12,18),(13,43),(14,44),(15,41),(16,42),(25,61),(26,62),(27,63),(28,64),(33,55),(34,56),(35,53),(36,54),(37,81),(38,82),(39,83),(40,84),(45,59),(46,60),(47,57),(48,58),(49,108),(50,105),(51,106),(52,107),(65,121),(66,122),(67,123),(68,124),(69,103),(70,104),(71,101),(72,102),(73,99),(74,100),(75,97),(76,98),(77,111),(78,112),(79,109),(80,110),(85,117),(86,118),(87,119),(88,120),(89,113),(90,114),(91,115),(92,116),(93,125),(94,126),(95,127),(96,128)], [(1,63),(2,64),(3,61),(4,62),(5,9),(6,10),(7,11),(8,12),(13,83),(14,84),(15,81),(16,82),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,57),(34,58),(35,59),(36,60),(37,41),(38,42),(39,43),(40,44),(45,53),(46,54),(47,55),(48,56),(49,78),(50,79),(51,80),(52,77),(65,93),(66,94),(67,95),(68,96),(69,73),(70,74),(71,75),(72,76),(85,89),(86,90),(87,91),(88,92),(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)], [(1,14),(2,15),(3,16),(4,13),(5,46),(6,47),(7,48),(8,45),(9,54),(10,55),(11,56),(12,53),(17,34),(18,35),(19,36),(20,33),(21,58),(22,59),(23,60),(24,57),(25,38),(26,39),(27,40),(28,37),(29,42),(30,43),(31,44),(32,41),(49,125),(50,126),(51,127),(52,128),(61,82),(62,83),(63,84),(64,81),(65,112),(66,109),(67,110),(68,111),(69,116),(70,113),(71,114),(72,115),(73,120),(74,117),(75,118),(76,119),(77,124),(78,121),(79,122),(80,123),(85,100),(86,97),(87,98),(88,99),(89,104),(90,101),(91,102),(92,103),(93,108),(94,105),(95,106),(96,107)], [(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)], [(1,106,18,113),(2,105,19,116),(3,108,20,115),(4,107,17,114),(5,88,28,79),(6,87,25,78),(7,86,26,77),(8,85,27,80),(9,92,32,50),(10,91,29,49),(11,90,30,52),(12,89,31,51),(13,96,34,71),(14,95,35,70),(15,94,36,69),(16,93,33,72),(21,118,62,111),(22,117,63,110),(23,120,64,109),(24,119,61,112),(37,122,46,99),(38,121,47,98),(39,124,48,97),(40,123,45,100),(41,126,54,103),(42,125,55,102),(43,128,56,101),(44,127,53,104),(57,76,82,65),(58,75,83,68),(59,74,84,67),(60,73,81,66)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 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 | 4 |
| 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 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,3,4,0,0,0,0,0,2],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,3,2] >;
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 | C23×C4⋊C4 | C22×C4⋊C4 | C24×C4 | C23×C4 | C24 | C24 |
| # reps | 1 | 28 | 3 | 32 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2^3\times C_4\rtimes C_4
% in TeX
G:=Group("C2^3xC4:C4"); // GroupNames label
G:=SmallGroup(128,2152);
// by ID
G=gap.SmallGroup(128,2152);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232]);
// 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,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀