C4^2.255C2^3 - GroupNames

G = C₄².₂₅₅C₂³ order 128 = 2⁷

116^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₂₅₅C₂³, C₄⋊C₄.₇₇D₄, (C₂×C₈).₁₉₃D₄, (C₂×Q₈).₆₄D₄, C₈⋊₄Q₈.₉C₂, C₈⋊₂C₈.₁₁C₂, C₄⋊C₈.₃₇C₂², C₄.Q₁₆.₈C₂, C₄⋊Q₈.₇₆C₂², C₄.₁₁₀(C₄○D₈), (C₄×C₈).₂₈₈C₂², Q₈⋊Q₈.₁₁C₂, C₄.₁₀D₈.₉C₂, C₄.₆Q₁₆.₈C₂, (C₄×Q₈).₅₂C₂², C₂.₁₁(C₈.D₄), C₄.₄₇(C₈.C₂²), C₄.SD₁₆.₁₃C₂, C₂.₂₃(D₄.₃D₄), C₂.₁₅(Q₈.D₄), C₂².₂₁₆(C₄⋊D₄), (C₂×C₄).₄₀(C₄○D₄), (C₂×C₄).₁₂₉₀(C₂×D₄), SmallGroup(128,436)

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

Derived series

C₁ — C₄² — C₄².₂₅₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₈⋊₄Q₈ — C₄².₂₅₅C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₂₅₅C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₂₅₅C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₂₅₅C₂³

Generators and relations for C₄².₂₅₅C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=a², d²=a^-1b², e²=b², ab=ba, cac^-1=a^-1, ad=da, ae=ea, cbc^-1=ebe^-1=b^-1, bd=db, dcd^-1=a^-1c, ece^-1=bc, ede^-1=a²d >

Subgroups: 144 in 70 conjugacy classes, 32 normal (all characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄×Q₈, C₄⋊Q₈, C₄.₁₀D₈, C₄.₆Q₁₆, C₈⋊₂C₈, C₈⋊₄Q₈, Q₈⋊Q₈, C₄.Q₁₆, C₄.SD₁₆, C₄².₂₅₅C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₄⋊D₄, C₄○D₈, C₈.C₂², Q₈.D₄, C₈.D₄, D₄.₃D₄, C₄².₂₅₅C₂³

Character table of C₄².₂₅₅C₂³

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	2	2	4	8	8	16	16	4	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	-1	-1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	1	1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	-1	-1	1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	-1	1	-1	1	1	1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	-1	-1	1	linear of order 2
ρ₉	2	2	2	2	-2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	-2	2	-2	-2	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	2	-2	-2	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	0	complex lifted from C₄○D₄
ρ₁₄	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2i	2i	0	√2	-√-2	-√2	0	0	√-2	complex lifted from C₄○D₈
ρ₁₆	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	2i	-2i	0	-√2	-√-2	√2	0	0	√-2	complex lifted from C₄○D₈
ρ₁₇	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	-2i	2i	0	-√2	√-2	√2	0	0	-√-2	complex lifted from C₄○D₈
ρ₁₈	2	-2	-2	2	-2	0	2	0	0	0	0	0	0	0	2i	-2i	0	√2	√-2	-√2	0	0	-√-2	complex lifted from C₄○D₈
ρ₁₉	4	-4	-4	4	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₀	4	-4	4	-4	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₁	4	-4	4	-4	0	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₂	4	4	-4	-4	0	0	0	0	0	0	0	0	0	2√-2	0	0	-2√-2	0	0	0	0	0	0	complex lifted from D₄.₃D₄
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	0	0	0	-2√-2	0	0	2√-2	0	0	0	0	0	0	complex lifted from D₄.₃D₄

Smallest permutation representation of C₄².₂₅₅C₂³
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

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

Matrix representation of C₄².₂₅₅C₂³ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	15	0	0
0	0	1	16	0	0
0	0	0	16	0	1
0	0	1	16	16	0

1	15	0	0	0	0
1	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	15	0	0	0	0
0	16	0	0	0	0
0	0	4	13	13	13
0	0	0	0	0	13
0	0	0	0	13	0
0	0	0	13	0	0

13	0	0	0	0	0
0	13	0	0	0	0
0	0	7	10	12	10
0	0	7	11	11	16
0	0	7	10	0	10
0	0	0	11	1	16

0	7	0	0	0	0
12	0	0	0	0	0
0	0	1	0	15	0
0	0	0	0	16	1
0	0	0	0	16	0
0	0	0	1	16	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,4,0,0,0,0,0,13,0,0,13,0,0,13,0,13,0,0,0,13,13,0,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,7,7,7,0,0,0,10,11,10,11,0,0,12,11,0,1,0,0,10,16,10,16],[0,12,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0] >;

C₄².₂₅₅C₂³ in GAP, Magma, Sage, TeX

C_4^2._{255}C_2^3

% in TeX

G:=Group("C4^2.255C2^3");

// GroupNames label

G:=SmallGroup(128,436);

// by ID

G=gap.SmallGroup(128,436);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,736,422,387,352,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄².₂₅₅C₂³ in TeX

G = C42.255C23 order 128 = 27

116th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₅₅C₂³ order 128 = 2⁷

116^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²