C4.Q32 - GroupNames

G = C₄.Q₃₂ order 128 = 2⁷

3^rd non-split extension by C₄ of Q₃₂ acting via Q₃₂/Q₁₆=C₂

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

Aliases: Q₁₆⋊₂Q₈, C₄.₇Q₃₂, C₄².₁₄₈D₄, (C₂×C₈).₇₂D₄, C₄⋊C₁₆.₁₀C₂, C₂.₇(C₂×Q₃₂), C₈.₃₂(C₂×Q₈), (C₂×C₄).₁₅₄D₈, (C₄×Q₁₆).₇C₂, C₁₆⋊₃C₄.₄C₂, C₈.₆₆(C₄○D₄), (C₂×C₁₆).₈C₂², (C₄×C₈).₆₈C₂², C₈⋊₂Q₈.₁₂C₂, (C₂×C₈).₅₂₉C₂³, C₂.Q₃₂.₃C₂, C₂².₁₁₅(C₂×D₈), C₄.₄₈(C₂²⋊Q₈), C₂.₁₅(C₁₆⋊C₂²), C₂.D₈.₁₄C₂², C₂.₁₅(D₄⋊Q₈), C₄.₁₁(C₈.C₂²), (C₂×Q₁₆).₁₁₁C₂², (C₂×C₄).₇₉₇(C₂×D₄), SmallGroup(128,959)

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

Derived series

C₁ — C₂×C₈ — C₄.Q₃₂

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×Q₁₆ — C₄×Q₁₆ — C₄.Q₃₂

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₄.Q₃₂

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — C₄.Q₃₂

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₄.Q₃₂

Generators and relations for C₄.Q₃₂
G = < a,b,c | a⁴=b¹⁶=1, c²=b⁸, bab^-1=a^-1, ac=ca, cbc^-1=a²b^-1 >

Subgroups: 148 in 65 conjugacy classes, 34 normal (22 characteristic)
C₁, C₂, C₄, C₄, C₄, C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, C₁₆, C₄², C₄², C₄⋊C₄, C₂×C₈, Q₁₆, Q₁₆, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₂.D₈, C₂.D₈, C₂.D₈, C₂×C₁₆, C₄×Q₈, C₄⋊Q₈, C₂×Q₁₆, C₂.Q₃₂, C₄⋊C₁₆, C₁₆⋊₃C₄, C₄×Q₁₆, C₈⋊₂Q₈, C₄.Q₃₂
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₈, C₂×D₄, C₂×Q₈, C₄○D₄, Q₃₂, C₂²⋊Q₈, C₂×D₈, C₈.C₂², D₄⋊Q₈, C₂×Q₃₂, C₁₆⋊C₂², C₄.Q₃₂

Character table of C₄.Q₃₂

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	2	2	2	2	4	8	8	8	8	16	16	2	2	2	2	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	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	-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	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	-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	-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	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	-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	1	1	1	1	1	1	linear of order 2
ρ₉	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₀	2	2	2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	√2	√2	-√2	√2	-√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₁	2	2	2	2	2	2	2	2	2	0	0	0	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-2	-2	2	-2	0	0	0	0	0	0	-2	-2	-2	-2	2	2	0	0	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	0	-√2	-√2	√2	-√2	√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	-2	2	2	0	0	-2	0	2	-2	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₆	2	-2	-2	2	2	0	0	-2	0	-2	2	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₁₈	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₁₉	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₀	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₁	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷-ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₂₂	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	symplectic lifted from Q₃₂, Schur index 2
ρ₂₃	2	2	-2	-2	0	-2	2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₄	2	2	-2	-2	0	2	-2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₅	2	-2	-2	2	2	0	0	-2	0	0	0	-2i	2i	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₆	2	-2	-2	2	2	0	0	-2	0	0	0	2i	-2i	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₇	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	2√2	-2√2	-2√2	2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²
ρ₂₈	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2√2	2√2	2√2	-2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²
ρ₂₉	4	-4	-4	4	-4	0	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of C₄.Q₃₂
►Regular action on 128 points

Generators in S₁₂₈

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

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

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

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

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

Matrix representation of C₄.Q₃₂ ►in GL₄(𝔽₁₇) generated by

16	0	0	0
0	16	0	0
0	0	6	9
0	0	11	11

11	4	0	0
13	11	0	0
0	0	15	3
0	0	16	2

12	12	0	0
12	5	0	0
0	0	7	2
0	0	10	10

G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,6,11,0,0,9,11],[11,13,0,0,4,11,0,0,0,0,15,16,0,0,3,2],[12,12,0,0,12,5,0,0,0,0,7,10,0,0,2,10] >;

C₄.Q₃₂ in GAP, Magma, Sage, TeX

C_4.Q_{32}

% in TeX

G:=Group("C4.Q32");

// GroupNames label

G:=SmallGroup(128,959);

// by ID

G=gap.SmallGroup(128,959);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,504,141,64,422,352,1684,438,242,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c|a^4=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;

// generators/relations

Export

Character table of C₄.Q₃₂ in TeX

G = C4.Q32 order 128 = 27

3rd non-split extension by C4 of Q32 acting via Q32/Q16=C2

G = C₄.Q₃₂ order 128 = 2⁷

3^rd non-split extension by C₄ of Q₃₂ acting via Q₃₂/Q₁₆=C₂