(C2xQ8):Q8 - GroupNames

G = (C₂×Q₈)⋊Q₈ order 128 = 2⁷

1^st semidirect product of C₂×Q₈ and Q₈ acting via Q₈/C₂=C₂²

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

Aliases: (C₂×Q₈)⋊₁Q₈, C₄⋊C₄.₈₃D₄, (C₂×C₄).₁₅Q₁₆, (C₂×C₄).₃₃SD₁₆, C₂.₆(Q₈⋊Q₈), C₂.₆(C₄.Q₁₆), (C₂²×C₄).₃₁₂D₄, C₂³.₉₀₆(C₂×D₄), C₄.₃₀(C₂²⋊Q₈), C₂².₅₄(C₂×Q₁₆), C₄.₃₁(C₄.₄D₄), (C₂²×C₈).₆₆C₂², C₂.₅(C₄.SD₁₆), C₂.₅(C₂³⋊Q₈), C₂².₉₄(C₂×SD₁₆), C₂².₂₀₈C₂²≀C₂, C₂.₂₀(C₂²⋊SD₁₆), C₂².₄Q₁₆.₂₁C₂, (C₂×C₄²).₃₅₄C₂², C₂.₂₀(C₂²⋊Q₁₆), (C₂²×Q₈).₆₀C₂², C₂².₁₃₃(C₈⋊C₂²), (C₂²×C₄).₁₄₄₀C₂³, C₂².₈₇(C₄.₄D₄), C₂².₁₀₃(C₂²⋊Q₈), C₂².₁₂₂(C₈.C₂²), C₂².₇C₄².₁₀C₂, C₂³.₆₇C₂³.₁₃C₂, C₂.₇(C₄².₂₈C₂²), (C₂×C₄⋊Q₈).₁₉C₂, (C₂×C₄).₂₈₀(C₂×Q₈), (C₂×C₄).₁₀₃₀(C₂×D₄), (C₂×C₄).₆₁₇(C₄○D₄), (C₂×C₄⋊C₄).₁₁₃C₂², (C₂×Q₈⋊C₄).₁₀C₂, SmallGroup(128,756)

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

Derived series

C₁ — C₂²×C₄ — (C₂×Q₈)⋊Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂³.₆₇C₂³ — (C₂×Q₈)⋊Q₈

Lower central

C₁ — C₂ — C₂²×C₄ — (C₂×Q₈)⋊Q₈

Upper central

C₁ — C₂³ — C₂×C₄² — (C₂×Q₈)⋊Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (C₂×Q₈)⋊Q₈

Generators and relations for (C₂×Q₈)⋊Q₈
G = < a,b,c,d,e | a²=b⁴=d⁴=1, c²=b², e²=d², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=ebe^-1=b^-1, bd=db, dcd^-1=ab²c, ece^-1=abc, ede^-1=d^-1 >

Subgroups: 304 in 145 conjugacy classes, 54 normal (28 characteristic)
C₁, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂.C₄², Q₈⋊C₄, C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊Q₈, C₂²×C₈, C₂²×Q₈, C₂²×Q₈, C₂².₇C₄², C₂².₄Q₁₆, C₂³.₆₇C₂³, C₂×Q₈⋊C₄, C₂×C₄⋊Q₈, (C₂×Q₈)⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, SD₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₂×SD₁₆, C₂×Q₁₆, C₈⋊C₂², C₈.C₂², C₂³⋊Q₈, C₂²⋊SD₁₆, C₂²⋊Q₁₆, Q₈⋊Q₈, C₄.Q₁₆, C₄.SD₁₆, C₄².₂₈C₂², (C₂×Q₈)⋊Q₈

Smallest permutation representation of (C₂×Q₈)⋊Q₈
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	-		-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	Q₈	SD₁₆	Q₁₆	C₄○D₄	C₈⋊C₂²	C₈.C₂²
kernel	(C₂×Q₈)⋊Q₈	C₂².₇C₄²	C₂².₄Q₁₆	C₂³.₆₇C₂³	C₂×Q₈⋊C₄	C₂×C₄⋊Q₈	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₂×C₄	C₂×C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	2	1	2	1	4	2	2	4	4	6	1	1

Matrix representation of (C₂×Q₈)⋊Q₈ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	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

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	16
0	0	0	0	1	0

1	15	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	12
0	0	0	0	12	5

7	5	0	0	0	0
7	10	0	0	0	0
0	0	10	2	0	0
0	0	9	7	0	0
0	0	0	0	0	1
0	0	0	0	16	0

4	9	0	0	0	0
0	13	0	0	0	0
0	0	4	0	0	0
0	0	11	13	0	0
0	0	0	0	4	0
0	0	0	0	0	13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,12,5],[7,7,0,0,0,0,5,10,0,0,0,0,0,0,10,9,0,0,0,0,2,7,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,4,11,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13] >;

(C₂×Q₈)⋊Q₈ in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes Q_8

% in TeX

G:=Group("(C2xQ8):Q8");

// GroupNames label

G:=SmallGroup(128,756);

// by ID

G=gap.SmallGroup(128,756);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,352,2804,718,172]);

// Polycyclic

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

// generators/relations

G = (C2×Q8)⋊Q8 order 128 = 27

1st semidirect product of C2×Q8 and Q8 acting via Q8/C2=C22

G = (C₂×Q₈)⋊Q₈ order 128 = 2⁷

1^st semidirect product of C₂×Q₈ and Q₈ acting via Q₈/C₂=C₂²