p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.175D4, C24.335C23, C23.466C24, C22.2502+ 1+4, (C2×D4).31Q8, C42⋊8C4⋊44C2, C42⋊9C4⋊29C2, C23.25(C2×Q8), C4.58(C22⋊Q8), C2.36(D4⋊3Q8), C23.Q8⋊31C2, C23.8Q8⋊69C2, (C23×C4).118C22, (C2×C42).567C22, (C22×C4).842C23, C22.317(C22×D4), C22.107(C22×Q8), (C22×D4).534C22, C2.25(C22.29C24), C24.3C22.49C2, C2.C42.202C22, C2.43(C22.26C24), C2.58(C22.47C24), (C4×C4⋊C4)⋊98C2, (C2×C4×D4).65C2, (C2×C4).360(C2×D4), (C2×C4).311(C2×Q8), C2.34(C2×C22⋊Q8), (C2×C42.C2)⋊14C2, (C2×C4).151(C4○D4), (C2×C4⋊C4).314C22, C22.342(C2×C4○D4), (C2×C22⋊C4).188C22, SmallGroup(128,1298)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.175D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 516 in 274 conjugacy classes, 112 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C42.C2, C23×C4, C22×D4, C4×C4⋊C4, C42⋊8C4, C42⋊9C4, C23.8Q8, C24.3C22, C23.Q8, C2×C4×D4, C2×C42.C2, C42.175D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C22.26C24, C22.29C24, C22.47C24, D4⋊3Q8, C42.175D4
►On 64 points
(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 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 18 38 46)(6 19 39 47)(7 20 40 48)(8 17 37 45)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(1 59 51 47)(2 58 52 46)(3 57 49 45)(4 60 50 48)(5 16 36 28)(6 15 33 27)(7 14 34 26)(8 13 35 25)(9 29 21 17)(10 32 22 20)(11 31 23 19)(12 30 24 18)(37 41 61 53)(38 44 62 56)(39 43 63 55)(40 42 64 54)
(1 40 9 5)(2 6 10 37)(3 38 11 7)(4 8 12 39)(13 18 43 48)(14 45 44 19)(15 20 41 46)(16 47 42 17)(21 36 51 64)(22 61 52 33)(23 34 49 62)(24 63 50 35)(25 30 55 60)(26 57 56 31)(27 32 53 58)(28 59 54 29)
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,59,51,47)(2,58,52,46)(3,57,49,45)(4,60,50,48)(5,16,36,28)(6,15,33,27)(7,14,34,26)(8,13,35,25)(9,29,21,17)(10,32,22,20)(11,31,23,19)(12,30,24,18)(37,41,61,53)(38,44,62,56)(39,43,63,55)(40,42,64,54), (1,40,9,5)(2,6,10,37)(3,38,11,7)(4,8,12,39)(13,18,43,48)(14,45,44,19)(15,20,41,46)(16,47,42,17)(21,36,51,64)(22,61,52,33)(23,34,49,62)(24,63,50,35)(25,30,55,60)(26,57,56,31)(27,32,53,58)(28,59,54,29)>;
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), (1,15,11,43)(2,16,12,44)(3,13,9,41)(4,14,10,42)(5,18,38,46)(6,19,39,47)(7,20,40,48)(8,17,37,45)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,35)(30,62,58,36)(31,63,59,33)(32,64,60,34), (1,59,51,47)(2,58,52,46)(3,57,49,45)(4,60,50,48)(5,16,36,28)(6,15,33,27)(7,14,34,26)(8,13,35,25)(9,29,21,17)(10,32,22,20)(11,31,23,19)(12,30,24,18)(37,41,61,53)(38,44,62,56)(39,43,63,55)(40,42,64,54), (1,40,9,5)(2,6,10,37)(3,38,11,7)(4,8,12,39)(13,18,43,48)(14,45,44,19)(15,20,41,46)(16,47,42,17)(21,36,51,64)(22,61,52,33)(23,34,49,62)(24,63,50,35)(25,30,55,60)(26,57,56,31)(27,32,53,58)(28,59,54,29) );
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)], [(1,15,11,43),(2,16,12,44),(3,13,9,41),(4,14,10,42),(5,18,38,46),(6,19,39,47),(7,20,40,48),(8,17,37,45),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,35),(30,62,58,36),(31,63,59,33),(32,64,60,34)], [(1,59,51,47),(2,58,52,46),(3,57,49,45),(4,60,50,48),(5,16,36,28),(6,15,33,27),(7,14,34,26),(8,13,35,25),(9,29,21,17),(10,32,22,20),(11,31,23,19),(12,30,24,18),(37,41,61,53),(38,44,62,56),(39,43,63,55),(40,42,64,54)], [(1,40,9,5),(2,6,10,37),(3,38,11,7),(4,8,12,39),(13,18,43,48),(14,45,44,19),(15,20,41,46),(16,47,42,17),(21,36,51,64),(22,61,52,33),(23,34,49,62),(24,63,50,35),(25,30,55,60),(26,57,56,31),(27,32,53,58),(28,59,54,29)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ 1+4 |
| kernel | C42.175D4 | C4×C4⋊C4 | C42⋊8C4 | C42⋊9C4 | C23.8Q8 | C24.3C22 | C23.Q8 | C2×C4×D4 | C2×C42.C2 | C42 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42.175D4 ►in GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 2 | 1 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,3,4,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,2,1,0,0,0,0,2,3],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,4,3,0,0,0,0,0,0,4,2,0,0,0,0,4,1] >;
C42.175D4 in GAP, Magma, Sage, TeX
C_4^2._{175}D_4 % in TeX
G:=Group("C4^2.175D4"); // GroupNames label
G:=SmallGroup(128,1298);
// by ID
G=gap.SmallGroup(128,1298);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,456,758,723,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations