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## G = C2×C2.Q32order 128 = 27

### Direct product of C2 and C2.Q32

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×C2.Q32
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C22×Q16 — C2×C2.Q32
 Lower central C1 — C2 — C4 — C8 — C2×C2.Q32
 Upper central C1 — C23 — C22×C4 — C22×C8 — C2×C2.Q32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C2.Q32

Generators and relations for C2×C2.Q32
G = < a,b,c,d | a2=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 244 in 116 conjugacy classes, 60 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×10], C23, C16 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×4], Q16 [×4], Q16 [×6], C22×C4, C22×C4 [×2], C2×Q8 [×9], C2.D8 [×2], C2.D8, C2×C16 [×2], C2×C16 [×2], C2×C4⋊C4, C22×C8, C2×Q16 [×6], C2×Q16 [×3], C22×Q8, C2.Q32 [×4], C2×C2.D8, C22×C16, C22×Q16, C2×C2.Q32
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], SD32 [×2], Q32 [×2], C2×C22⋊C4, C2×D8, C2×SD16, C2.Q32 [×4], C2×D4⋊C4, C2×SD32, C2×Q32, C2×C2.Q32

Smallest permutation representation of C2×C2.Q32
Regular action on 128 points
Generators in S128
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 124)(18 125)(19 126)(20 127)(21 128)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 121)(31 122)(32 123)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 65)(45 66)(46 67)(47 68)(48 69)(49 94)(50 95)(51 96)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 120)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 128)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 97)(89 98)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)
(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)
(1 125 9 117)(2 37 10 45)(3 123 11 115)(4 35 12 43)(5 121 13 113)(6 33 14 41)(7 119 15 127)(8 47 16 39)(17 87 25 95)(18 111 26 103)(19 85 27 93)(20 109 28 101)(21 83 29 91)(22 107 30 99)(23 81 31 89)(24 105 32 97)(34 61 42 53)(36 59 44 51)(38 57 46 49)(40 55 48 63)(50 124 58 116)(52 122 60 114)(54 120 62 128)(56 118 64 126)(65 96 73 88)(66 104 74 112)(67 94 75 86)(68 102 76 110)(69 92 77 84)(70 100 78 108)(71 90 79 82)(72 98 80 106)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 8A ··· 8H 16A ··· 16P order 1 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 16 ··· 16 size 1 1 ··· 1 2 2 2 2 8 ··· 8 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + - image C1 C2 C2 C2 C2 C4 D4 D4 D8 SD16 D8 SD32 Q32 kernel C2×C2.Q32 C2.Q32 C2×C2.D8 C22×C16 C22×Q16 C2×Q16 C2×C8 C22×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 3 1 2 4 2 8 8

Matrix representation of C2×C2.Q32 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 1 0 0 0 0 13 11 0 0 6 13
,
 16 0 0 0 0 16 0 0 0 0 5 12 0 0 12 12
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,13,6,0,0,11,13],[16,0,0,0,0,16,0,0,0,0,5,12,0,0,12,12] >;

C2×C2.Q32 in GAP, Magma, Sage, TeX

C_2\times C_2.Q_{32}
% in TeX

G:=Group("C2xC2.Q32");
// GroupNames label

G:=SmallGroup(128,869);
// by ID

G=gap.SmallGroup(128,869);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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