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## G = C42.30Q8order 128 = 27

### 30th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.30Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.30Q8
 Lower central C1 — C2 — C2×C4 — C42.30Q8
 Upper central C1 — C23 — C2×C42 — C42.30Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.30Q8

Generators and relations for C42.30Q8
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 236 in 128 conjugacy classes, 72 normal (36 characteristic)
C1, C2 [×7], C4 [×4], C4 [×4], C4 [×8], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42, C4⋊C8 [×4], C4.Q8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×2], C4×C4⋊C4, C429C4, C2×C4⋊C8, C2×C4.Q8 [×2], C42.30Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C4.Q8 [×4], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16 [×2], C8⋊C22, C8.C22, C23.65C23, C2×C4.Q8, M4(2)⋊C4, C4⋊SD16, D4.D4, Q8⋊Q8, D42Q8, C42.30Q8

Smallest permutation representation of C42.30Q8
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 39 29 13)(2 40 30 14)(3 37 31 15)(4 38 32 16)(5 19 108 120)(6 20 105 117)(7 17 106 118)(8 18 107 119)(9 49 35 25)(10 50 36 26)(11 51 33 27)(12 52 34 28)(21 126 122 110)(22 127 123 111)(23 128 124 112)(24 125 121 109)(41 57 71 81)(42 58 72 82)(43 59 69 83)(44 60 70 84)(45 61 67 53)(46 62 68 54)(47 63 65 55)(48 64 66 56)(73 103 113 85)(74 104 114 86)(75 101 115 87)(76 102 116 88)(77 89 93 99)(78 90 94 100)(79 91 95 97)(80 92 96 98)
(1 71 35 53)(2 72 36 54)(3 69 33 55)(4 70 34 56)(5 96 124 88)(6 93 121 85)(7 94 122 86)(8 95 123 87)(9 61 29 41)(10 62 30 42)(11 63 31 43)(12 64 32 44)(13 81 49 45)(14 82 50 46)(15 83 51 47)(16 84 52 48)(17 90 110 114)(18 91 111 115)(19 92 112 116)(20 89 109 113)(21 104 106 78)(22 101 107 79)(23 102 108 80)(24 103 105 77)(25 67 39 57)(26 68 40 58)(27 65 37 59)(28 66 38 60)(73 117 99 125)(74 118 100 126)(75 119 97 127)(76 120 98 128)
(1 87 9 79)(2 86 10 78)(3 85 11 77)(4 88 12 80)(5 84 23 66)(6 83 24 65)(7 82 21 68)(8 81 22 67)(13 75 25 91)(14 74 26 90)(15 73 27 89)(16 76 28 92)(17 72 126 62)(18 71 127 61)(19 70 128 64)(20 69 125 63)(29 101 35 95)(30 104 36 94)(31 103 33 93)(32 102 34 96)(37 113 51 99)(38 116 52 98)(39 115 49 97)(40 114 50 100)(41 111 53 119)(42 110 54 118)(43 109 55 117)(44 112 56 120)(45 107 57 123)(46 106 58 122)(47 105 59 121)(48 108 60 124)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + - + - + + - image C1 C2 C2 C2 C2 C2 C4 Q8 D4 Q8 D4 SD16 C4○D4 C8⋊C22 C8.C22 kernel C42.30Q8 C22.4Q16 C4×C4⋊C4 C42⋊9C4 C2×C4⋊C8 C2×C4.Q8 C4⋊C8 C42 C4⋊C4 C4⋊C4 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 2 1 1 1 2 8 2 2 2 2 8 4 1 1

Matrix representation of C42.30Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 2 0 0 0 0 16 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 13 6 0 0 0 0 6 4 0 0 0 0 0 0 13 9 0 0 0 0 4 4
,
 10 1 0 0 0 0 1 7 0 0 0 0 0 0 10 16 0 0 0 0 16 7 0 0 0 0 0 0 16 0 0 0 0 0 1 1

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,6,0,0,0,0,6,4,0,0,0,0,0,0,13,4,0,0,0,0,9,4],[10,1,0,0,0,0,1,7,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,16,1,0,0,0,0,0,1] >;`

C42.30Q8 in GAP, Magma, Sage, TeX

`C_4^2._{30}Q_8`
`% in TeX`

`G:=Group("C4^2.30Q8");`
`// GroupNames label`

`G:=SmallGroup(128,680);`
`// by ID`

`G=gap.SmallGroup(128,680);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,100,2019,1018,248]);`
`// Polycyclic`

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

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