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G = C4214Q8order 128 = 27

1st semidirect product of C42 and Q8 acting via Q8/C4=C2

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

Aliases: C4214Q8, C43.12C2, C23.177C24, (C4×Q8)⋊17C4, C4.49(C4×Q8), C42.175(C2×C4), C424C4.11C2, C22.68(C23×C4), C4.13(C42⋊C2), C22.25(C22×Q8), C4(C23.67C23), (C2×C42).1005C22, (C22×C4).1239C23, (C22×Q8).390C22, C42(C23.65C23), C42(C23.63C23), C2.C42.512C22, C23.65C23.93C2, C23.63C23.66C2, C23.67C23.67C2, C2.2(C23.37C23), C2.4(C23.36C23), C2.7(C2×C4×Q8), C2.7(C4×C4○D4), (C4×C4⋊C4).31C2, (C2×C4×Q8).19C2, C4⋊C4.199(C2×C4), (C2×C4).160(C2×Q8), (C2×Q8).191(C2×C4), C22.69(C2×C4○D4), (C2×C4).636(C4○D4), (C2×C4⋊C4).792C22, (C2×C4).210(C22×C4), C2.17(C2×C42⋊C2), SmallGroup(128,1027)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4214Q8
C1C2C22C23C22×C4C2×C42C43 — C4214Q8
C1C22 — C4214Q8
C1C22×C4 — C4214Q8
C1C23 — C4214Q8

Generators and relations for C4214Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 348 in 250 conjugacy classes, 160 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×30], C2×C4 [×30], Q8 [×8], C23, C42 [×16], C42 [×14], C4⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×4], C2.C42 [×14], C2×C42 [×3], C2×C42 [×8], C2×C4⋊C4, C2×C4⋊C4 [×8], C4×Q8 [×8], C22×Q8, C43, C424C4 [×2], C4×C4⋊C4, C4×C4⋊C4 [×2], C23.63C23 [×4], C23.65C23 [×2], C23.67C23 [×2], C2×C4×Q8, C4214Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×10], C24, C42⋊C2 [×4], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4 [×5], C2×C42⋊C2, C2×C4×Q8, C4×C4○D4, C23.36C23 [×2], C23.37C23 [×2], C4214Q8

Smallest permutation representation of C4214Q8
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 89 125 121)(34 90 126 122)(35 91 127 123)(36 92 128 124)(65 74 78 102)(66 75 79 103)(67 76 80 104)(68 73 77 101)(81 117 113 93)(82 118 114 94)(83 119 115 95)(84 120 116 96)(85 109 105 98)(86 110 106 99)(87 111 107 100)(88 112 108 97)
(1 13 5 59)(2 14 6 60)(3 15 7 57)(4 16 8 58)(9 63 55 17)(10 64 56 18)(11 61 53 19)(12 62 54 20)(21 47 39 29)(22 48 40 30)(23 45 37 31)(24 46 38 32)(25 51 43 69)(26 52 44 70)(27 49 41 71)(28 50 42 72)(33 75 100 93)(34 76 97 94)(35 73 98 95)(36 74 99 96)(65 106 116 124)(66 107 113 121)(67 108 114 122)(68 105 115 123)(77 85 83 91)(78 86 84 92)(79 87 81 89)(80 88 82 90)(101 109 119 127)(102 110 120 128)(103 111 117 125)(104 112 118 126)
(1 104 5 118)(2 73 6 95)(3 102 7 120)(4 75 8 93)(9 82 55 80)(10 115 56 68)(11 84 53 78)(12 113 54 66)(13 126 59 112)(14 35 60 98)(15 128 57 110)(16 33 58 100)(17 90 63 88)(18 123 64 105)(19 92 61 86)(20 121 62 107)(21 74 39 96)(22 103 40 117)(23 76 37 94)(24 101 38 119)(25 65 43 116)(26 79 44 81)(27 67 41 114)(28 77 42 83)(29 99 47 36)(30 111 48 125)(31 97 45 34)(32 109 46 127)(49 122 71 108)(50 91 72 85)(51 124 69 106)(52 89 70 87)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4AF4AG···4AV
order12···24···44···44···4
size11···11···12···24···4

56 irreducible representations

dim11111111122
type++++++++-
imageC1C2C2C2C2C2C2C2C4Q8C4○D4
kernelC4214Q8C43C424C4C4×C4⋊C4C23.63C23C23.65C23C23.67C23C2×C4×Q8C4×Q8C42C2×C4
# reps1123422116420

Matrix representation of C4214Q8 in GL5(𝔽5)

30000
02000
00300
00020
00002
,
10000
02000
00200
00040
00004
,
10000
02000
00300
00020
00003
,
40000
00400
01000
00003
00030

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

C4214Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{14}Q_8
% in TeX

G:=Group("C4^2:14Q8");
// GroupNames label

G:=SmallGroup(128,1027);
// by ID

G=gap.SmallGroup(128,1027);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,184,80]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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