direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C42.C2, C22.26C25, C24.657C23, C42.737C23, C23.269C24, C2.4(Q8×C23), (C2×C4).30C24, C4.18(C22×Q8), C4⋊C4.455C23, (C22×C4).105Q8, C23.149(C2×Q8), (C22×C42).36C2, C23.381(C4○D4), C22.50(C22×Q8), (C23×C4).579C22, (C22×C4).1173C23, (C2×C42).1139C22, (C2×C4).250(C2×Q8), (C22×C4⋊C4).47C2, C2.10(C22×C4○D4), (C2×C4⋊C4).943C22, C22.150(C2×C4○D4), SmallGroup(128,2169)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
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
Generators and relations
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 >
►Regular action on 128 points
Generators in S128
(1 29)(2 30)(3 31)(4 32)(5 63)(6 64)(7 61)(8 62)(9 17)(10 18)(11 19)(12 20)(13 60)(14 57)(15 58)(16 59)(21 47)(22 48)(23 45)(24 46)(25 51)(26 52)(27 49)(28 50)(33 56)(34 53)(35 54)(36 55)(37 73)(38 74)(39 75)(40 76)(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 27)(2 28)(3 25)(4 26)(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 39)(2 60 56 40)(3 57 53 37)(4 58 54 38)(5 25 21 17)(6 26 22 18)(7 27 23 19)(8 28 24 20)(9 63 51 47)(10 64 52 48)(11 61 49 45)(12 62 50 46)(13 33 76 30)(14 34 73 31)(15 35 74 32)(16 36 75 29)(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 70 21 43)(6 44 22 71)(7 72 23 41)(8 42 24 69)(9 98 51 119)(10 120 52 99)(11 100 49 117)(12 118 50 97)(13 88 76 91)(14 92 73 85)(15 86 74 89)(16 90 75 87)(17 95 25 66)(18 67 26 96)(19 93 27 68)(20 65 28 94)(29 83 36 78)(30 79 33 84)(31 81 34 80)(32 77 35 82)(37 108 57 125)(38 126 58 105)(39 106 59 127)(40 128 60 107)(45 112 61 121)(46 122 62 109)(47 110 63 123)(48 124 64 111)
G:=sub<Sym(128)| (1,29)(2,30)(3,31)(4,32)(5,63)(6,64)(7,61)(8,62)(9,17)(10,18)(11,19)(12,20)(13,60)(14,57)(15,58)(16,59)(21,47)(22,48)(23,45)(24,46)(25,51)(26,52)(27,49)(28,50)(33,56)(34,53)(35,54)(36,55)(37,73)(38,74)(39,75)(40,76)(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), (1,27)(2,28)(3,25)(4,26)(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), (1,59,55,39)(2,60,56,40)(3,57,53,37)(4,58,54,38)(5,25,21,17)(6,26,22,18)(7,27,23,19)(8,28,24,20)(9,63,51,47)(10,64,52,48)(11,61,49,45)(12,62,50,46)(13,33,76,30)(14,34,73,31)(15,35,74,32)(16,36,75,29)(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)(5,70,21,43)(6,44,22,71)(7,72,23,41)(8,42,24,69)(9,98,51,119)(10,120,52,99)(11,100,49,117)(12,118,50,97)(13,88,76,91)(14,92,73,85)(15,86,74,89)(16,90,75,87)(17,95,25,66)(18,67,26,96)(19,93,27,68)(20,65,28,94)(29,83,36,78)(30,79,33,84)(31,81,34,80)(32,77,35,82)(37,108,57,125)(38,126,58,105)(39,106,59,127)(40,128,60,107)(45,112,61,121)(46,122,62,109)(47,110,63,123)(48,124,64,111)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,63)(6,64)(7,61)(8,62)(9,17)(10,18)(11,19)(12,20)(13,60)(14,57)(15,58)(16,59)(21,47)(22,48)(23,45)(24,46)(25,51)(26,52)(27,49)(28,50)(33,56)(34,53)(35,54)(36,55)(37,73)(38,74)(39,75)(40,76)(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), (1,27)(2,28)(3,25)(4,26)(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), (1,59,55,39)(2,60,56,40)(3,57,53,37)(4,58,54,38)(5,25,21,17)(6,26,22,18)(7,27,23,19)(8,28,24,20)(9,63,51,47)(10,64,52,48)(11,61,49,45)(12,62,50,46)(13,33,76,30)(14,34,73,31)(15,35,74,32)(16,36,75,29)(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)(5,70,21,43)(6,44,22,71)(7,72,23,41)(8,42,24,69)(9,98,51,119)(10,120,52,99)(11,100,49,117)(12,118,50,97)(13,88,76,91)(14,92,73,85)(15,86,74,89)(16,90,75,87)(17,95,25,66)(18,67,26,96)(19,93,27,68)(20,65,28,94)(29,83,36,78)(30,79,33,84)(31,81,34,80)(32,77,35,82)(37,108,57,125)(38,126,58,105)(39,106,59,127)(40,128,60,107)(45,112,61,121)(46,122,62,109)(47,110,63,123)(48,124,64,111) );
G=PermutationGroup([(1,29),(2,30),(3,31),(4,32),(5,63),(6,64),(7,61),(8,62),(9,17),(10,18),(11,19),(12,20),(13,60),(14,57),(15,58),(16,59),(21,47),(22,48),(23,45),(24,46),(25,51),(26,52),(27,49),(28,50),(33,56),(34,53),(35,54),(36,55),(37,73),(38,74),(39,75),(40,76),(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)], [(1,27),(2,28),(3,25),(4,26),(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)], [(1,59,55,39),(2,60,56,40),(3,57,53,37),(4,58,54,38),(5,25,21,17),(6,26,22,18),(7,27,23,19),(8,28,24,20),(9,63,51,47),(10,64,52,48),(11,61,49,45),(12,62,50,46),(13,33,76,30),(14,34,73,31),(15,35,74,32),(16,36,75,29),(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),(5,70,21,43),(6,44,22,71),(7,72,23,41),(8,42,24,69),(9,98,51,119),(10,120,52,99),(11,100,49,117),(12,118,50,97),(13,88,76,91),(14,92,73,85),(15,86,74,89),(16,90,75,87),(17,95,25,66),(18,67,26,96),(19,93,27,68),(20,65,28,94),(29,83,36,78),(30,79,33,84),(31,81,34,80),(32,77,35,82),(37,108,57,125),(38,126,58,105),(39,106,59,127),(40,128,60,107),(45,112,61,121),(46,122,62,109),(47,110,63,123),(48,124,64,111)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀