p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.6C23, C8⋊Q8⋊1C2, C4⋊C4.41D4, (C2×D4).31D4, (C2×Q8).31D4, C8.2D4.2C2, C4⋊Q8.36C22, C8⋊C4.1C22, C2.28(D4⋊4D4), C22.187C22≀C2, C42.C2.4C22, C42.2C22⋊3C2, C4.4D4.10C22, C2.20(D4.10D4), C42.C22.1C2, C22.57C24.1C2, (C2×C4).219(C2×D4), SmallGroup(128,392)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.6C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=e2=a2, ab=ba, cac-1=dad=a-1, eae-1=a-1b2, cbc-1=ebe-1=b-1, dbd=a2b-1, dcd=ac, ece-1=bc, de=ed >
Subgroups: 224 in 98 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C8⋊C4, C4.Q8, C2.D8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C2×SD16, C2×Q16, C42.C22, C42.2C22, C8.2D4, C8⋊Q8, C22.57C24, C42.6C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4⋊4D4, D4.10D4, C42.6C23
Character table of C42.6C23
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ16 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ17 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ18 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ19 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ20 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | symplectic lifted from D4.10D4, Schur index 2 |
►On 64 points
Generators in S64
(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)
(1 20 13 5)(2 17 14 6)(3 18 15 7)(4 19 16 8)(9 62 54 57)(10 63 55 58)(11 64 56 59)(12 61 53 60)(21 29 35 26)(22 30 36 27)(23 31 33 28)(24 32 34 25)(37 44 47 52)(38 41 48 49)(39 42 45 50)(40 43 46 51)
(1 39 3 37)(2 38 4 40)(5 42 7 44)(6 41 8 43)(9 36 11 34)(10 35 12 33)(13 45 15 47)(14 48 16 46)(17 49 19 51)(18 52 20 50)(21 53 23 55)(22 56 24 54)(25 57 27 59)(26 60 28 58)(29 61 31 63)(30 64 32 62)
(2 4)(5 18)(6 17)(7 20)(8 19)(9 53)(10 56)(11 55)(12 54)(14 16)(21 33)(22 36)(23 35)(24 34)(25 27)(30 32)(37 40)(38 39)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(57 58)(59 60)(61 64)(62 63)
(1 29 3 31)(2 25 4 27)(5 35 7 33)(6 24 8 22)(9 46 11 48)(10 39 12 37)(13 26 15 28)(14 32 16 30)(17 34 19 36)(18 23 20 21)(38 54 40 56)(41 62 43 64)(42 60 44 58)(45 53 47 55)(49 57 51 59)(50 61 52 63)
G:=sub<Sym(64)| (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), (1,20,13,5)(2,17,14,6)(3,18,15,7)(4,19,16,8)(9,62,54,57)(10,63,55,58)(11,64,56,59)(12,61,53,60)(21,29,35,26)(22,30,36,27)(23,31,33,28)(24,32,34,25)(37,44,47,52)(38,41,48,49)(39,42,45,50)(40,43,46,51), (1,39,3,37)(2,38,4,40)(5,42,7,44)(6,41,8,43)(9,36,11,34)(10,35,12,33)(13,45,15,47)(14,48,16,46)(17,49,19,51)(18,52,20,50)(21,53,23,55)(22,56,24,54)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62), (2,4)(5,18)(6,17)(7,20)(8,19)(9,53)(10,56)(11,55)(12,54)(14,16)(21,33)(22,36)(23,35)(24,34)(25,27)(30,32)(37,40)(38,39)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,58)(59,60)(61,64)(62,63), (1,29,3,31)(2,25,4,27)(5,35,7,33)(6,24,8,22)(9,46,11,48)(10,39,12,37)(13,26,15,28)(14,32,16,30)(17,34,19,36)(18,23,20,21)(38,54,40,56)(41,62,43,64)(42,60,44,58)(45,53,47,55)(49,57,51,59)(50,61,52,63)>;
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), (1,20,13,5)(2,17,14,6)(3,18,15,7)(4,19,16,8)(9,62,54,57)(10,63,55,58)(11,64,56,59)(12,61,53,60)(21,29,35,26)(22,30,36,27)(23,31,33,28)(24,32,34,25)(37,44,47,52)(38,41,48,49)(39,42,45,50)(40,43,46,51), (1,39,3,37)(2,38,4,40)(5,42,7,44)(6,41,8,43)(9,36,11,34)(10,35,12,33)(13,45,15,47)(14,48,16,46)(17,49,19,51)(18,52,20,50)(21,53,23,55)(22,56,24,54)(25,57,27,59)(26,60,28,58)(29,61,31,63)(30,64,32,62), (2,4)(5,18)(6,17)(7,20)(8,19)(9,53)(10,56)(11,55)(12,54)(14,16)(21,33)(22,36)(23,35)(24,34)(25,27)(30,32)(37,40)(38,39)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,58)(59,60)(61,64)(62,63), (1,29,3,31)(2,25,4,27)(5,35,7,33)(6,24,8,22)(9,46,11,48)(10,39,12,37)(13,26,15,28)(14,32,16,30)(17,34,19,36)(18,23,20,21)(38,54,40,56)(41,62,43,64)(42,60,44,58)(45,53,47,55)(49,57,51,59)(50,61,52,63) );
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)], [(1,20,13,5),(2,17,14,6),(3,18,15,7),(4,19,16,8),(9,62,54,57),(10,63,55,58),(11,64,56,59),(12,61,53,60),(21,29,35,26),(22,30,36,27),(23,31,33,28),(24,32,34,25),(37,44,47,52),(38,41,48,49),(39,42,45,50),(40,43,46,51)], [(1,39,3,37),(2,38,4,40),(5,42,7,44),(6,41,8,43),(9,36,11,34),(10,35,12,33),(13,45,15,47),(14,48,16,46),(17,49,19,51),(18,52,20,50),(21,53,23,55),(22,56,24,54),(25,57,27,59),(26,60,28,58),(29,61,31,63),(30,64,32,62)], [(2,4),(5,18),(6,17),(7,20),(8,19),(9,53),(10,56),(11,55),(12,54),(14,16),(21,33),(22,36),(23,35),(24,34),(25,27),(30,32),(37,40),(38,39),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(57,58),(59,60),(61,64),(62,63)], [(1,29,3,31),(2,25,4,27),(5,35,7,33),(6,24,8,22),(9,46,11,48),(10,39,12,37),(13,26,15,28),(14,32,16,30),(17,34,19,36),(18,23,20,21),(38,54,40,56),(41,62,43,64),(42,60,44,58),(45,53,47,55),(49,57,51,59),(50,61,52,63)]])
Matrix representation of C42.6C23 ►in GL8(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 | 0 | 0 |
| 6 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 1 | 7 | 16 |
| 0 | 0 | 0 | 0 | 7 | 6 | 6 | 10 |
| 0 | 0 | 0 | 0 | 7 | 16 | 11 | 1 |
| 0 | 0 | 0 | 0 | 6 | 10 | 7 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 16 | 4 | 6 | 10 |
| 0 | 0 | 0 | 0 | 10 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 11 | 7 | 1 | 13 |
G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,15,4],[0,0,6,11,0,0,0,0,0,0,11,11,0,0,0,0,7,10,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,0,0,11,7,7,6,0,0,0,0,1,6,16,10,0,0,0,0,7,6,11,7,0,0,0,0,16,10,1,6],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,16],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,13,16,10,11,0,0,0,0,0,4,0,7,0,0,0,0,7,6,4,1,0,0,0,0,0,10,0,13] >;
C42.6C23 in GAP, Magma, Sage, TeX
C_4^2._6C_2^3
% in TeX
G:=Group("C4^2.6C2^3"); // GroupNames label
G:=SmallGroup(128,392);
// by ID
G=gap.SmallGroup(128,392);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,184,1123,570,521,136,3924,1411,998,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=e^2=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,e*a*e^-1=a^-1*b^2,c*b*c^-1=e*b*e^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=a*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations
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