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## G = C42⋊7Q8order 128 = 27

### 7th semidirect product of C42 and Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42⋊7Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42⋊7Q8
 Lower central C1 — C23 — C42⋊7Q8
 Upper central C1 — C23 — C42⋊7Q8
 Jennings C1 — C23 — C42⋊7Q8

Generators and relations for C427Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 468 in 278 conjugacy classes, 140 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×30], C2×C4 [×30], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×16], C4⋊C4 [×26], C22×C4, C22×C4 [×14], C2×Q8 [×20], C2.C42 [×8], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×18], C4⋊Q8 [×16], C22×Q8 [×4], C4×C4⋊C4 [×2], C429C4, C23.65C23 [×4], C23.78C23 [×4], C2×C4⋊Q8 [×4], C427Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×12], C23 [×15], C2×D4 [×12], C2×Q8 [×18], C24, C4⋊Q8 [×8], C22×D4 [×2], C22×Q8 [×3], 2+ 1+4, 2- 1+4, C2×C4⋊Q8 [×2], C23.41C23, D42, D4×Q8 [×2], Q82, C427Q8

Smallest permutation representation of C427Q8
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 20 45 13)(2 17 46 14)(3 18 47 15)(4 19 48 16)(5 37 127 36)(6 38 128 33)(7 39 125 34)(8 40 126 35)(9 42 23 50)(10 43 24 51)(11 44 21 52)(12 41 22 49)(25 54 30 57)(26 55 31 58)(27 56 32 59)(28 53 29 60)(61 65 100 72)(62 66 97 69)(63 67 98 70)(64 68 99 71)(73 108 111 86)(74 105 112 87)(75 106 109 88)(76 107 110 85)(77 113 82 102)(78 114 83 103)(79 115 84 104)(80 116 81 101)(89 117 96 124)(90 118 93 121)(91 119 94 122)(92 120 95 123)
(1 98 58 52)(2 62 59 43)(3 100 60 50)(4 64 57 41)(5 101 109 122)(6 115 110 118)(7 103 111 124)(8 113 112 120)(9 18 72 28)(10 14 69 32)(11 20 70 26)(12 16 71 30)(13 67 31 21)(15 65 29 23)(17 66 27 24)(19 68 25 22)(33 79 107 90)(34 83 108 96)(35 77 105 92)(36 81 106 94)(37 80 88 91)(38 84 85 93)(39 78 86 89)(40 82 87 95)(42 47 61 53)(44 45 63 55)(46 97 56 51)(48 99 54 49)(73 117 125 114)(74 123 126 102)(75 119 127 116)(76 121 128 104)
(1 119 58 116)(2 123 59 102)(3 117 60 114)(4 121 57 104)(5 63 109 44)(6 99 110 49)(7 61 111 42)(8 97 112 51)(9 39 72 86)(10 35 69 105)(11 37 70 88)(12 33 71 107)(13 94 31 81)(14 92 32 77)(15 96 29 83)(16 90 30 79)(17 95 27 82)(18 89 28 78)(19 93 25 84)(20 91 26 80)(21 36 67 106)(22 38 68 85)(23 34 65 108)(24 40 66 87)(41 128 64 76)(43 126 62 74)(45 122 55 101)(46 120 56 113)(47 124 53 103)(48 118 54 115)(50 125 100 73)(52 127 98 75)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4Z 4AA 4AB 4AC 4AD order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 Q8 D4 Q8 2+ 1+4 2- 1+4 kernel C42⋊7Q8 C4×C4⋊C4 C42⋊9C4 C23.65C23 C23.78C23 C2×C4⋊Q8 C42 C4⋊C4 C4⋊C4 C22 C22 # reps 1 2 1 4 4 4 4 8 8 1 1

Matrix representation of C427Q8 in GL6(𝔽5)

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

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

C427Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_7Q_8
% in TeX

G:=Group("C4^2:7Q8");
// GroupNames label

G:=SmallGroup(128,1283);
// by ID

G=gap.SmallGroup(128,1283);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,184,675,80]);
// Polycyclic

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

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