Copied to
clipboard

G = D76⋊C3order 456 = 23·3·19

The semidirect product of D76 and C3 acting faithfully

metacyclic, supersoluble, monomial

Aliases: D76⋊C3, C761C6, D381C6, C4⋊(C19⋊C6), C19⋊C31D4, C191(C3×D4), C38.3(C2×C6), (C2×C19⋊C6)⋊1C2, (C4×C19⋊C3)⋊1C2, C2.4(C2×C19⋊C6), (C2×C19⋊C3).3C22, SmallGroup(456,9)

Series: Derived Chief Lower central Upper central

C1C38 — D76⋊C3
C1C19C38C2×C19⋊C3C2×C19⋊C6 — D76⋊C3
C19C38 — D76⋊C3
C1C2C4

Generators and relations for D76⋊C3
 G = < a,b,c | a76=b2=c3=1, bab=a-1, cac-1=a49, cbc-1=a48b >

38C2
38C2
19C3
19C22
19C22
19C6
38C6
38C6
2D19
2D19
19D4
19C2×C6
19C2×C6
19C12
2C19⋊C6
2C19⋊C6
19C3×D4

Character table of D76⋊C3

 class 12A2B2C3A3B46A6B6C6D6E6F12A12B19A19B19C38A38B38C76A76B76C76D76E76F
 size 113838191921919383838383838666666666666
ρ1111111111111111111111111111    trivial
ρ2111-111-1111-11-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ311-1111-111-11-11-1-1111111-1-1-1-1-1-1    linear of order 2
ρ411-1-111111-1-1-1-111111111111111    linear of order 2
ρ5111-1ζ32ζ3-1ζ32ζ3ζ32ζ65ζ3ζ6ζ65ζ6111111-1-1-1-1-1-1    linear of order 6
ρ611-1-1ζ3ζ321ζ3ζ32ζ65ζ6ζ6ζ65ζ32ζ3111111111111    linear of order 6
ρ711-11ζ32ζ3-1ζ32ζ3ζ6ζ3ζ65ζ32ζ65ζ6111111-1-1-1-1-1-1    linear of order 6
ρ8111-1ζ3ζ32-1ζ3ζ32ζ3ζ6ζ32ζ65ζ6ζ65111111-1-1-1-1-1-1    linear of order 6
ρ91111ζ32ζ31ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ32111111111111    linear of order 3
ρ101111ζ3ζ321ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ3111111111111    linear of order 3
ρ1111-1-1ζ32ζ31ζ32ζ3ζ6ζ65ζ65ζ6ζ3ζ32111111111111    linear of order 6
ρ1211-11ζ3ζ32-1ζ3ζ32ζ65ζ32ζ6ζ3ζ6ζ65111111-1-1-1-1-1-1    linear of order 6
ρ132-200220-2-2000000222-2-2-2000000    orthogonal lifted from D4
ρ142-200-1--3-1+-301+-31--3000000222-2-2-2000000    complex lifted from C3×D4
ρ152-200-1+-3-1--301--31+-3000000222-2-2-2000000    complex lifted from C3×D4
ρ16660000600000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191719161914195193192ζ19181912191119819719ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192    orthogonal lifted from C19⋊C6
ρ17660000-600000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191519131910199196194ζ19181912191119819719ζ1917191619141951931921915191319101991961941917191619141951931921918191219111981971919151913191019919619419181912191119819719191719161914195193192    orthogonal lifted from C2×C19⋊C6
ρ18660000-600000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ19181912191119819719ζ191719161914195193192ζ1915191319101991961941918191219111981971919151913191019919619419171916191419519319219181912191119819719191719161914195193192191519131910199196194    orthogonal lifted from C2×C19⋊C6
ρ19660000-600000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191719161914195193192ζ191519131910199196194ζ191819121911198197191917191619141951931921918191219111981971919151913191019919619419171916191419519319219151913191019919619419181912191119819719    orthogonal lifted from C2×C19⋊C6
ρ20660000600000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ19181912191119819719ζ191519131910199196194ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719    orthogonal lifted from C19⋊C6
ρ21660000600000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191519131910199196194ζ191719161914195193192ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194    orthogonal lifted from C19⋊C6
ρ226-60000000000000ζ19181912191119819719ζ191719161914195193192ζ19151913191019919619419171916191419519319219151913191019919619419181912191119819719ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ1944ζ19184ζ19124ζ19114ζ1984ζ1974ζ19    orthogonal faithful
ρ236-60000000000000ζ191519131910199196194ζ19181912191119819719ζ19171916191419519319219181912191119819719191719161914195193192191519131910199196194ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ1924ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194    orthogonal faithful
ρ246-60000000000000ζ191719161914195193192ζ191519131910199196194ζ1918191219111981971919151913191019919619419181912191119819719191719161914195193192ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ1944ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192    orthogonal faithful
ρ256-60000000000000ζ19181912191119819719ζ191719161914195193192ζ19151913191019919619419171916191419519319219151913191019919619419181912191119819719ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ1924ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19    orthogonal faithful
ρ266-60000000000000ζ191719161914195193192ζ191519131910199196194ζ1918191219111981971919151913191019919619419181912191119819719191719161914195193192ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ1924ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192    orthogonal faithful
ρ276-60000000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192191819121911198197191917191619141951931921915191319101991961944ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194    orthogonal faithful

Smallest permutation representation of D76⋊C3
On 76 points
Generators in S76
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)
(2 46 50)(3 15 23)(4 60 72)(5 29 45)(6 74 18)(7 43 67)(8 12 40)(9 57 13)(10 26 62)(11 71 35)(14 54 30)(16 68 52)(17 37 25)(19 51 47)(21 65 69)(22 34 42)(24 48 64)(27 31 59)(28 76 32)(33 73 49)(36 56 44)(38 70 66)(41 53 61)(55 75 63)

G:=sub<Sym(76)| (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49), (2,46,50)(3,15,23)(4,60,72)(5,29,45)(6,74,18)(7,43,67)(8,12,40)(9,57,13)(10,26,62)(11,71,35)(14,54,30)(16,68,52)(17,37,25)(19,51,47)(21,65,69)(22,34,42)(24,48,64)(27,31,59)(28,76,32)(33,73,49)(36,56,44)(38,70,66)(41,53,61)(55,75,63)>;

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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49), (2,46,50)(3,15,23)(4,60,72)(5,29,45)(6,74,18)(7,43,67)(8,12,40)(9,57,13)(10,26,62)(11,71,35)(14,54,30)(16,68,52)(17,37,25)(19,51,47)(21,65,69)(22,34,42)(24,48,64)(27,31,59)(28,76,32)(33,73,49)(36,56,44)(38,70,66)(41,53,61)(55,75,63) );

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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,76),(21,75),(22,74),(23,73),(24,72),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49)], [(2,46,50),(3,15,23),(4,60,72),(5,29,45),(6,74,18),(7,43,67),(8,12,40),(9,57,13),(10,26,62),(11,71,35),(14,54,30),(16,68,52),(17,37,25),(19,51,47),(21,65,69),(22,34,42),(24,48,64),(27,31,59),(28,76,32),(33,73,49),(36,56,44),(38,70,66),(41,53,61),(55,75,63)]])

Matrix representation of D76⋊C3 in GL8(𝔽229)

183000000
160228000000
001004077607920
00209221881492081
0022822812110019227
0021903068131210
0019991419919228
00100000
,
2280000000
691000000
001004077607920
0010813019113288129
001211201717120121
0012988132191130108
002079607740100
00120814918822209
,
1340000000
0134000000
00100000
0012988132191130108
00000100
001004077607920
0022819991419919
0021903068131210

G:=sub<GL(8,GF(229))| [1,160,0,0,0,0,0,0,83,228,0,0,0,0,0,0,0,0,100,209,228,2,19,1,0,0,40,22,228,190,99,0,0,0,77,188,121,30,141,0,0,0,60,149,100,68,99,0,0,0,79,208,19,131,19,0,0,0,20,1,227,210,228,0],[228,69,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,100,108,121,129,20,1,0,0,40,130,120,88,79,208,0,0,77,191,17,132,60,149,0,0,60,132,17,191,77,188,0,0,79,88,120,130,40,22,0,0,20,129,121,108,100,209],[134,0,0,0,0,0,0,0,0,134,0,0,0,0,0,0,0,0,1,129,0,100,228,2,0,0,0,88,0,40,19,190,0,0,0,132,0,77,99,30,0,0,0,191,1,60,141,68,0,0,0,130,0,79,99,131,0,0,0,108,0,20,19,210] >;

D76⋊C3 in GAP, Magma, Sage, TeX

D_{76}\rtimes C_3
% in TeX

G:=Group("D76:C3");
// GroupNames label

G:=SmallGroup(456,9);
// by ID

G=gap.SmallGroup(456,9);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,141,66,10804,1064]);
// Polycyclic

G:=Group<a,b,c|a^76=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^48*b>;
// generators/relations

Export

Subgroup lattice of D76⋊C3 in TeX
Character table of D76⋊C3 in TeX

׿
×
𝔽