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G = C4×Q8⋊C4order 128 = 27

Direct product of C4 and Q8⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4×Q8⋊C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C4×Q8⋊C4
 Lower central C1 — C2 — C4 — C4×Q8⋊C4
 Upper central C1 — C22×C4 — C2×C42 — C4×Q8⋊C4
 Jennings C1 — C2 — C2 — C22×C4 — C4×Q8⋊C4

Generators and relations for C4×Q8⋊C4
G = < a,b,c,d | a4=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 268 in 166 conjugacy classes, 92 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C4×C8, Q8⋊C4, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×C8, C22×Q8, C22.4Q16, C4×C4⋊C4, C2×C4×C8, C2×Q8⋊C4, C2×C4×Q8, C4×Q8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4○D4, Q8⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C2×SD16, C2×Q16, C4○D8, C4×C22⋊C4, C2×Q8⋊C4, C23.24D4, C4×SD16, C4×Q16, C4×Q8⋊C4

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 SD16 Q16 C4○D4 C4○D8 kernel C4×Q8⋊C4 C22.4Q16 C4×C4⋊C4 C2×C4×C8 C2×Q8⋊C4 C2×C4×Q8 Q8⋊C4 C4×Q8 C42 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 1 2 1 16 8 2 2 4 4 4 8

Matrix representation of C4×Q8⋊C4 in GL5(𝔽17)

 16 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 16 0
,
 1 0 0 0 0 0 4 6 0 0 0 6 13 0 0 0 0 0 12 12 0 0 0 12 5
,
 4 0 0 0 0 0 10 16 0 0 0 16 7 0 0 0 0 0 13 0 0 0 0 0 4

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,4,6,0,0,0,6,13,0,0,0,0,0,12,12,0,0,0,12,5],[4,0,0,0,0,0,10,16,0,0,0,16,7,0,0,0,0,0,13,0,0,0,0,0,4] >;

C4×Q8⋊C4 in GAP, Magma, Sage, TeX

C_4\times Q_8\rtimes C_4
% in TeX

G:=Group("C4xQ8:C4");
// GroupNames label

G:=SmallGroup(128,493);
// by ID

G=gap.SmallGroup(128,493);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,436,2019,248,4037,124]);
// Polycyclic

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

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