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## G = C42⋊15Q8order 128 = 27

### 2nd semidirect product of C42 and Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42⋊15Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C42⋊15Q8
 Lower central C1 — C23 — C42⋊15Q8
 Upper central C1 — C23 — C42⋊15Q8
 Jennings C1 — C23 — C42⋊15Q8

Generators and relations for C4215Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 324 in 204 conjugacy classes, 108 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×20], C22, C22 [×6], C2×C4 [×18], C2×C4 [×36], Q8 [×4], C23, C42 [×4], C42 [×12], C4⋊C4 [×18], C22×C4, C22×C4 [×14], C2×Q8 [×6], C2.C42 [×18], C2×C42, C2×C42 [×6], C2×C4⋊C4 [×9], C22×Q8, C43, C425C4 [×2], C23.63C23 [×6], C23.65C23 [×3], C23.67C23 [×3], C4215Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×12], C24, C422C2 [×4], C22×Q8, C2×C4○D4 [×6], C2×C422C2, C23.36C23 [×3], C23.37C23 [×3], C4215Q8

Smallest permutation representation of C4215Q8
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 41 55 16)(2 42 56 13)(3 43 53 14)(4 44 54 15)(5 112 39 85)(6 109 40 86)(7 110 37 87)(8 111 38 88)(9 29 18 58)(10 30 19 59)(11 31 20 60)(12 32 17 57)(21 98 46 65)(22 99 47 66)(23 100 48 67)(24 97 45 68)(25 69 50 64)(26 70 51 61)(27 71 52 62)(28 72 49 63)(33 114 126 81)(34 115 127 82)(35 116 128 83)(36 113 125 84)(73 95 106 118)(74 96 107 119)(75 93 108 120)(76 94 105 117)(77 122 104 89)(78 123 101 90)(79 124 102 91)(80 121 103 92)
(1 21 18 72)(2 22 19 69)(3 23 20 70)(4 24 17 71)(5 76 84 122)(6 73 81 123)(7 74 82 124)(8 75 83 121)(9 63 55 46)(10 64 56 47)(11 61 53 48)(12 62 54 45)(13 66 30 25)(14 67 31 26)(15 68 32 27)(16 65 29 28)(33 101 109 95)(34 102 110 96)(35 103 111 93)(36 104 112 94)(37 107 115 91)(38 108 116 92)(39 105 113 89)(40 106 114 90)(41 98 58 49)(42 99 59 50)(43 100 60 51)(44 97 57 52)(77 85 117 125)(78 86 118 126)(79 87 119 127)(80 88 120 128)
(1 79 18 119)(2 103 19 93)(3 77 20 117)(4 101 17 95)(5 65 84 28)(6 99 81 50)(7 67 82 26)(8 97 83 52)(9 96 55 102)(10 120 56 80)(11 94 53 104)(12 118 54 78)(13 90 30 106)(14 124 31 74)(15 92 32 108)(16 122 29 76)(21 127 72 87)(22 35 69 111)(23 125 70 85)(24 33 71 109)(25 40 66 114)(27 38 68 116)(34 63 110 46)(36 61 112 48)(37 100 115 51)(39 98 113 49)(41 89 58 105)(42 123 59 73)(43 91 60 107)(44 121 57 75)(45 126 62 86)(47 128 64 88)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4AB 4AC ··· 4AJ order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C2 Q8 C4○D4 kernel C42⋊15Q8 C43 C42⋊5C4 C23.63C23 C23.65C23 C23.67C23 C42 C2×C4 # reps 1 1 2 6 3 3 4 24

Matrix representation of C4215Q8 in GL6(𝔽5)

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

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

C4215Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{15}Q_8
% in TeX

G:=Group("C4^2:15Q8");
// GroupNames label

G:=SmallGroup(128,1595);
// by ID

G=gap.SmallGroup(128,1595);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,184,2019,80]);
// Polycyclic

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

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