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G = C42⋊8Q8  order 128 = 27

8th semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C42⋊8Q8, C23.505C24, C22.2082- 1+4, C22.2852+ 1+4, C42⋊4C4.22C2, C42⋊5C4.10C2, (C2×C42).592C22, (C22×C4).124C23, C22.126(C22×Q8), (C22×Q8).147C22, C2.70(C22.45C24), C23.63C23.34C2, C2.C42.235C22, C23.78C23.13C2, C23.67C23.47C2, C23.83C23.21C2, C2.16(C23.41C23), C2.37(C23.37C23), C2.53(C22.50C24), (C4×C4⋊C4).76C2, (C2×C4).127(C2×Q8), (C2×C4).163(C4○D4), (C2×C4⋊C4).344C22, C22.381(C2×C4○D4), SmallGroup(128,1337)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C42⋊8Q8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊4C4 — C42⋊8Q8
Lower central
C1 — C23 — C42⋊8Q8
Upper central
C1 — C23 — C42⋊8Q8
Jennings
C1 — C23 — C42⋊8Q8

Generators and relations for C42⋊8Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 340 in 198 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C22×Q8, C42⋊4C4, C4×C4⋊C4, C42⋊5C4, C23.63C23, C23.67C23, C23.78C23, C23.83C23, C42⋊8Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.37C23, C23.41C23, C22.45C24, C22.50C24, C42⋊8Q8

Smallest permutation representation of C42⋊8Q8
►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 57 25 17)(2 58 26 18)(3 59 27 19)(4 60 28 20)(5 35 77 82)(6 36 78 83)(7 33 79 84)(8 34 80 81)(9 50 100 72)(10 51 97 69)(11 52 98 70)(12 49 99 71)(13 29 53 47)(14 30 54 48)(15 31 55 45)(16 32 56 46)(21 65 61 43)(22 66 62 44)(23 67 63 41)(24 68 64 42)(37 126 103 116)(38 127 104 113)(39 128 101 114)(40 125 102 115)(73 123 93 109)(74 124 94 110)(75 121 95 111)(76 122 96 112)(85 105 89 119)(86 106 90 120)(87 107 91 117)(88 108 92 118)

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4Z4AA4AB4AC4AD
order12···24···44···44444
size11···12···24···48888

38 irreducible representations

dim111111112244
type++++++++-+-
imageC1C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC42⋊8Q8C42⋊4C4C4×C4⋊C4C42⋊5C4C23.63C23C23.67C23C23.78C23C23.83C23C42C2×C4C22C22
# reps1111442241611

Matrix representation of C42⋊8Q8 ►in GL6(𝔽5)

020000
200000
000400
004000
000040
000004
,
010000
100000
003000
000300
000040
000004
,
100000
010000
002000
000300
000020
000043
,
030000
200000
000200
002000
000023
000003

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

C42⋊8Q8 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_8Q_8
 
% in TeX
 
G:=Group("C4^2:8Q8");
 
// GroupNames label
 
G:=SmallGroup(128,1337);
 
// by ID
 
G=gap.SmallGroup(128,1337);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,184,675,248]);
 
// 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=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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