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## G = C2×Q8⋊3Q8order 128 = 27

### Direct product of C2 and Q8⋊3Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×Q8⋊3Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×Q8 — C2×Q8⋊3Q8
 Lower central C1 — C22 — C2×Q8⋊3Q8
 Upper central C1 — C23 — C2×Q8⋊3Q8
 Jennings C1 — C22 — C2×Q8⋊3Q8

Generators and relations for C2×Q83Q8
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b2c, ede-1=d-1 >

Subgroups: 604 in 524 conjugacy classes, 444 normal (11 characteristic)
C1, C2 [×3], C2 [×4], C4 [×16], C4 [×22], C22, C22 [×6], C2×C4 [×46], C2×C4 [×22], Q8 [×16], Q8 [×24], C23, C42 [×36], C4⋊C4 [×88], C22×C4, C22×C4 [×14], C2×Q8 [×24], C2×Q8 [×12], C2×C42 [×9], C2×C4⋊C4, C2×C4⋊C4 [×21], C4×Q8 [×48], C42.C2 [×48], C4⋊Q8 [×24], C22×Q8, C22×Q8 [×3], C2×C4×Q8, C2×C4×Q8 [×5], C2×C42.C2 [×6], C2×C4⋊Q8 [×3], Q83Q8 [×16], C2×Q83Q8
Quotients: C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C4○D4 [×4], C24 [×31], C22×Q8 [×14], C2×C4○D4 [×6], 2- 1+4 [×2], C25, Q83Q8 [×4], Q8×C23, C22×C4○D4, C2×2- 1+4, C2×Q83Q8

Smallest permutation representation of C2×Q83Q8
Regular action on 128 points
Generators in S128
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 113)(14 114)(15 115)(16 116)(17 38)(18 39)(19 40)(20 37)(25 32)(26 29)(27 30)(28 31)(33 54)(34 55)(35 56)(36 53)(41 46)(42 47)(43 48)(44 45)(49 72)(50 69)(51 70)(52 71)(57 62)(58 63)(59 64)(60 61)(65 88)(66 85)(67 86)(68 87)(73 80)(74 77)(75 78)(76 79)(81 102)(82 103)(83 104)(84 101)(89 96)(90 93)(91 94)(92 95)(97 118)(98 119)(99 120)(100 117)(105 110)(106 111)(107 112)(108 109)(121 126)(122 127)(123 128)(124 125)
(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 33 3 35)(2 36 4 34)(5 56 7 54)(6 55 8 53)(9 42 11 44)(10 41 12 43)(13 83 15 81)(14 82 16 84)(17 70 19 72)(18 69 20 71)(21 46 23 48)(22 45 24 47)(25 64 27 62)(26 63 28 61)(29 58 31 60)(30 57 32 59)(37 52 39 50)(38 51 40 49)(65 118 67 120)(66 117 68 119)(73 110 75 112)(74 109 76 111)(77 108 79 106)(78 107 80 105)(85 100 87 98)(86 99 88 97)(89 126 91 128)(90 125 92 127)(93 124 95 122)(94 123 96 121)(101 114 103 116)(102 113 104 115)
(1 17 24 31)(2 18 21 32)(3 19 22 29)(4 20 23 30)(5 40 11 26)(6 37 12 27)(7 38 9 28)(8 39 10 25)(13 99 122 109)(14 100 123 110)(15 97 124 111)(16 98 121 112)(33 70 47 60)(34 71 48 57)(35 72 45 58)(36 69 46 59)(41 64 53 50)(42 61 54 51)(43 62 55 52)(44 63 56 49)(65 90 79 104)(66 91 80 101)(67 92 77 102)(68 89 78 103)(73 84 85 94)(74 81 86 95)(75 82 87 96)(76 83 88 93)(105 114 117 128)(106 115 118 125)(107 116 119 126)(108 113 120 127)
(1 65 24 79)(2 66 21 80)(3 67 22 77)(4 68 23 78)(5 86 11 74)(6 87 12 75)(7 88 9 76)(8 85 10 73)(13 61 122 51)(14 62 123 52)(15 63 124 49)(16 64 121 50)(17 104 31 90)(18 101 32 91)(19 102 29 92)(20 103 30 89)(25 94 39 84)(26 95 40 81)(27 96 37 82)(28 93 38 83)(33 120 47 108)(34 117 48 105)(35 118 45 106)(36 119 46 107)(41 112 53 98)(42 109 54 99)(43 110 55 100)(44 111 56 97)(57 128 71 114)(58 125 72 115)(59 126 69 116)(60 127 70 113)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 4Y ··· 4AP order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 2 2 4 type + + + + + - - image C1 C2 C2 C2 C2 Q8 C4○D4 2- 1+4 kernel C2×Q8⋊3Q8 C2×C4×Q8 C2×C42.C2 C2×C4⋊Q8 Q8⋊3Q8 C2×Q8 C2×C4 C22 # reps 1 6 6 3 16 8 8 2

Matrix representation of C2×Q83Q8 in GL5(𝔽5)

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

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

C2×Q83Q8 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3Q_8
% in TeX

G:=Group("C2xQ8:3Q8");
// GroupNames label

G:=SmallGroup(128,2208);
// by ID

G=gap.SmallGroup(128,2208);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,1430,352,570,136]);
// Polycyclic

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

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