Copied to
clipboard

## G = C4⋊C4.Q8order 128 = 27

### 7th non-split extension by C4⋊C4 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C4⋊C4.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22.4Q16 — C4⋊C4.Q8
 Lower central C1 — C2 — C22×C4 — C4⋊C4.Q8
 Upper central C1 — C23 — C2×C42 — C4⋊C4.Q8
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.Q8

Generators and relations for C4⋊C4.Q8
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b-1, dbd-1=a-1b-1, dcd-1=a2c-1 >

Subgroups: 208 in 99 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22.7C42, C22.4Q16, C428C4, C23.65C23, C4⋊C4.Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C4○D8, C8⋊C22, C8.C22, C23.83C23, D4.Q8, Q8.Q8, C23.19D4, C23.20D4, C42.78C22, C42.28C22, C4⋊C4.Q8

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

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 type + + + + + - + + - image C1 C2 C2 C2 C2 Q8 D4 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel C4⋊C4.Q8 C22.7C42 C22.4Q16 C42⋊8C4 C23.65C23 C4⋊C4 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 4 1 1 2 2 10 8 1 1

Matrix representation of C4⋊C4.Q8 in GL6(𝔽17)

 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 0 1 0 0 0 0 16 0
,
 5 12 0 0 0 0 12 12 0 0 0 0 0 0 5 14 0 0 0 0 3 12 0 0 0 0 0 0 4 6 0 0 0 0 6 13
,
 5 5 0 0 0 0 5 12 0 0 0 0 0 0 14 5 0 0 0 0 12 3 0 0 0 0 0 0 0 13 0 0 0 0 4 0
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 5 3 0 0 0 0 14 12 0 0 0 0 0 0 13 6 0 0 0 0 6 4

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,5,3,0,0,0,0,14,12,0,0,0,0,0,0,4,6,0,0,0,0,6,13],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,14,12,0,0,0,0,5,3,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,13,6,0,0,0,0,6,4] >;

C4⋊C4.Q8 in GAP, Magma, Sage, TeX

C_4\rtimes C_4.Q_8
% in TeX

G:=Group("C4:C4.Q8");
// GroupNames label

G:=SmallGroup(128,833);
// by ID

G=gap.SmallGroup(128,833);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,504,141,176,422,387,58,2028,1027,124]);
// Polycyclic

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

׿
×
𝔽