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G = C424Q8order 128 = 27

4th semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C424Q8, C23.213C24, C22.512+ 1+4, C22.352- 1+4, C4⋊Q831C4, C4.20(C4×Q8), C42.182(C2×C4), C424C4.15C2, C428C4.21C2, C22.35(C22×Q8), C22.104(C23×C4), (C2×C42).420C22, (C22×C4).478C23, (C22×Q8).88C22, C2.18(C22.11C24), C23.63C23.4C2, C2.C42.49C22, C23.65C23.31C2, C23.67C23.26C2, C2.5(C23.41C23), C2.7(C22.36C24), C2.10(C23.32C23), C2.13(C2×C4×Q8), (C4×C4⋊C4).36C2, (C2×C4⋊Q8).25C2, C4⋊C4.106(C2×C4), (C2×C4).163(C2×Q8), (C2×C4).33(C22×C4), (C2×Q8).110(C2×C4), C22.98(C2×C4○D4), (C2×C4).652(C4○D4), (C2×C4⋊C4).183C22, SmallGroup(128,1063)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C424Q8
C1C2C22C23C22×C4C2×C42C424C4 — C424Q8
C1C22 — C424Q8
C1C23 — C424Q8
C1C23 — C424Q8

Generators and relations for C424Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 364 in 230 conjugacy classes, 148 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×22], C2×C4 [×34], Q8 [×8], C23, C42 [×8], C42 [×6], C4⋊C4 [×16], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×8], C2×Q8 [×4], C2.C42 [×16], C2×C42 [×3], C2×C42 [×4], C2×C4⋊C4 [×10], C4⋊Q8 [×8], C22×Q8 [×2], C424C4, C4×C4⋊C4 [×2], C428C4, C23.63C23 [×4], C23.65C23 [×2], C23.67C23 [×4], C2×C4⋊Q8, C424Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, 2+ 1+4 [×2], 2- 1+4 [×2], C2×C4×Q8, C22.11C24, C23.32C23, C22.36C24 [×2], C23.41C23 [×2], C424Q8

Smallest permutation representation of C424Q8
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 27 23 55)(2 28 24 56)(3 25 21 53)(4 26 22 54)(5 41 37 9)(6 42 38 10)(7 43 39 11)(8 44 40 12)(13 49 45 17)(14 50 46 18)(15 51 47 19)(16 52 48 20)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 121 125 89)(34 122 126 90)(35 123 127 91)(36 124 128 92)(65 102 78 74)(66 103 79 75)(67 104 80 76)(68 101 77 73)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)(85 98 105 109)(86 99 106 110)(87 100 107 111)(88 97 108 112)
(1 45 37 59)(2 14 38 32)(3 47 39 57)(4 16 40 30)(5 31 23 13)(6 60 24 46)(7 29 21 15)(8 58 22 48)(9 71 27 17)(10 64 28 50)(11 69 25 19)(12 62 26 52)(18 42 72 56)(20 44 70 54)(33 75 111 117)(34 104 112 94)(35 73 109 119)(36 102 110 96)(41 63 55 49)(43 61 53 51)(65 106 84 92)(66 87 81 121)(67 108 82 90)(68 85 83 123)(74 99 120 128)(76 97 118 126)(77 105 115 91)(78 86 116 124)(79 107 113 89)(80 88 114 122)(93 125 103 100)(95 127 101 98)
(1 80 37 114)(2 68 38 83)(3 78 39 116)(4 66 40 81)(5 82 23 67)(6 115 24 77)(7 84 21 65)(8 113 22 79)(9 94 27 104)(10 119 28 73)(11 96 25 102)(12 117 26 75)(13 90 31 108)(14 123 32 85)(15 92 29 106)(16 121 30 87)(17 34 71 112)(18 127 72 98)(19 36 69 110)(20 125 70 100)(33 62 111 52)(35 64 109 50)(41 118 55 76)(42 95 56 101)(43 120 53 74)(44 93 54 103)(45 122 59 88)(46 91 60 105)(47 124 57 86)(48 89 58 107)(49 126 63 97)(51 128 61 99)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4AJ
order12···24···44···4
size11···12···24···4

44 irreducible representations

dim1111111112244
type++++++++-+-
imageC1C2C2C2C2C2C2C2C4Q8C4○D42+ 1+42- 1+4
kernelC424Q8C424C4C4×C4⋊C4C428C4C23.63C23C23.65C23C23.67C23C2×C4⋊Q8C4⋊Q8C42C2×C4C22C22
# reps11214241164422

Matrix representation of C424Q8 in GL8(𝔽5)

20000000
02000000
00400000
00040000
00000001
00002423
00000413
00004000
,
10000000
01000000
00400000
00040000
00000100
00004000
00004320
00002423
,
44000000
21000000
00230000
00030000
00000010
00004320
00004000
00000142
,
30000000
42000000
00140000
00240000
00003024
00004033
00002112
00004231

G:=sub<GL(8,GF(5))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,1,3,3,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,1,0,3,4,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,3],[4,2,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,3,0,1,0,0,0,0,1,2,0,4,0,0,0,0,0,0,0,2],[3,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,3,4,2,4,0,0,0,0,0,0,1,2,0,0,0,0,2,3,1,3,0,0,0,0,4,3,2,1] >;

C424Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4Q_8
% in TeX

G:=Group("C4^2:4Q8");
// GroupNames label

G:=SmallGroup(128,1063);
// by ID

G=gap.SmallGroup(128,1063);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,219,268,675,80]);
// 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=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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