p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.30C24, C22.69C25, C42.566C23, Q8⋊10(C4○D4), C4○2(Q8⋊3Q8), C4○2(Q8⋊5D4), C4○2(Q8⋊6D4), Q8⋊6D4⋊33C2, Q8⋊5D4⋊37C2, Q8⋊3Q8⋊33C2, C4⋊C4.478C23, (C2×C4).600C24, Q8○2(C42⋊C2), C4⋊Q8.336C22, (C2×D4).461C23, (C4×D4).355C22, C22⋊C4.91C23, (C4×Q8).323C22, (C2×Q8).437C23, C4⋊1D4.182C22, C4⋊D4.220C22, (C2×C42).937C22, C22⋊Q8.226C22, C2.13(C2.C25), C4○(C22.53C24), C42⋊2C2.14C22, (C22×C4).1203C23, C22.26C24⋊30C2, C4.4D4.171C22, C42.C2.149C22, (C22×Q8).497C22, C4○2(C22.50C24), C4○2(C22.47C24), C22.50C24⋊45C2, C22.53C24⋊28C2, C22.47C24⋊44C2, C23.36C23⋊22C2, C42⋊C2.342C22, C22.D4.27C22, (C2×C4×Q8)⋊57C2, (C4×C4○D4)⋊22C2, (C2×C4)⋊21(C4○D4), C42⋊C2○(C4×Q8), (C2×C4)○(Q8⋊5D4), (C2×C4)○(Q8⋊3Q8), C4.177(C2×C4○D4), (C2×Q8)○(C42⋊C2), C2.40(C22×C4○D4), C22.34(C2×C4○D4), (C2×C4⋊C4).961C22, (C2×C4○D4).325C22, (C2×C4)○(C22.50C24), SmallGroup(128,2212)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.69C25
G = < a,b,c,d,e,f,g | a2=b2=f2=1, c2=d2=e2=b, g2=a, ab=ba, dcd-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, cg=gc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 756 in 556 conjugacy classes, 398 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C22×Q8, C2×C4○D4, C2×C4×Q8, C4×C4○D4, C23.36C23, C22.26C24, Q8⋊5D4, Q8⋊6D4, C22.47C24, C22.50C24, Q8⋊3Q8, C22.53C24, C22.69C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, C25, C22×C4○D4, C2.C25, C22.69C25
►On 64 points
(1 10)(2 11)(3 12)(4 9)(5 27)(6 28)(7 25)(8 26)(13 43)(14 44)(15 41)(16 42)(17 62)(18 63)(19 64)(20 61)(21 59)(22 60)(23 57)(24 58)(29 45)(30 46)(31 47)(32 48)(33 53)(34 54)(35 55)(36 56)(37 51)(38 52)(39 49)(40 50)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 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 32 3 30)(2 45 4 47)(5 21 7 23)(6 60 8 58)(9 31 11 29)(10 48 12 46)(13 61 15 63)(14 17 16 19)(18 43 20 41)(22 26 24 28)(25 57 27 59)(33 37 35 39)(34 52 36 50)(38 56 40 54)(42 64 44 62)(49 53 51 55)
(1 26 3 28)(2 25 4 27)(5 11 7 9)(6 10 8 12)(13 53 15 55)(14 56 16 54)(17 40 19 38)(18 39 20 37)(21 29 23 31)(22 32 24 30)(33 41 35 43)(34 44 36 42)(45 57 47 59)(46 60 48 58)(49 61 51 63)(50 64 52 62)
(1 54)(2 55)(3 56)(4 53)(5 41)(6 42)(7 43)(8 44)(9 33)(10 34)(11 35)(12 36)(13 25)(14 26)(15 27)(16 28)(17 58)(18 59)(19 60)(20 57)(21 63)(22 64)(23 61)(24 62)(29 49)(30 50)(31 51)(32 52)(37 47)(38 48)(39 45)(40 46)
(1 23 10 57)(2 24 11 58)(3 21 12 59)(4 22 9 60)(5 48 27 32)(6 45 28 29)(7 46 25 30)(8 47 26 31)(13 50 43 40)(14 51 44 37)(15 52 41 38)(16 49 42 39)(17 55 62 35)(18 56 63 36)(19 53 64 33)(20 54 61 34)
G:=sub<Sym(64)| (1,10)(2,11)(3,12)(4,9)(5,27)(6,28)(7,25)(8,26)(13,43)(14,44)(15,41)(16,42)(17,62)(18,63)(19,64)(20,61)(21,59)(22,60)(23,57)(24,58)(29,45)(30,46)(31,47)(32,48)(33,53)(34,54)(35,55)(36,56)(37,51)(38,52)(39,49)(40,50), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (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), (1,32,3,30)(2,45,4,47)(5,21,7,23)(6,60,8,58)(9,31,11,29)(10,48,12,46)(13,61,15,63)(14,17,16,19)(18,43,20,41)(22,26,24,28)(25,57,27,59)(33,37,35,39)(34,52,36,50)(38,56,40,54)(42,64,44,62)(49,53,51,55), (1,26,3,28)(2,25,4,27)(5,11,7,9)(6,10,8,12)(13,53,15,55)(14,56,16,54)(17,40,19,38)(18,39,20,37)(21,29,23,31)(22,32,24,30)(33,41,35,43)(34,44,36,42)(45,57,47,59)(46,60,48,58)(49,61,51,63)(50,64,52,62), (1,54)(2,55)(3,56)(4,53)(5,41)(6,42)(7,43)(8,44)(9,33)(10,34)(11,35)(12,36)(13,25)(14,26)(15,27)(16,28)(17,58)(18,59)(19,60)(20,57)(21,63)(22,64)(23,61)(24,62)(29,49)(30,50)(31,51)(32,52)(37,47)(38,48)(39,45)(40,46), (1,23,10,57)(2,24,11,58)(3,21,12,59)(4,22,9,60)(5,48,27,32)(6,45,28,29)(7,46,25,30)(8,47,26,31)(13,50,43,40)(14,51,44,37)(15,52,41,38)(16,49,42,39)(17,55,62,35)(18,56,63,36)(19,53,64,33)(20,54,61,34)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,27)(6,28)(7,25)(8,26)(13,43)(14,44)(15,41)(16,42)(17,62)(18,63)(19,64)(20,61)(21,59)(22,60)(23,57)(24,58)(29,45)(30,46)(31,47)(32,48)(33,53)(34,54)(35,55)(36,56)(37,51)(38,52)(39,49)(40,50), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (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), (1,32,3,30)(2,45,4,47)(5,21,7,23)(6,60,8,58)(9,31,11,29)(10,48,12,46)(13,61,15,63)(14,17,16,19)(18,43,20,41)(22,26,24,28)(25,57,27,59)(33,37,35,39)(34,52,36,50)(38,56,40,54)(42,64,44,62)(49,53,51,55), (1,26,3,28)(2,25,4,27)(5,11,7,9)(6,10,8,12)(13,53,15,55)(14,56,16,54)(17,40,19,38)(18,39,20,37)(21,29,23,31)(22,32,24,30)(33,41,35,43)(34,44,36,42)(45,57,47,59)(46,60,48,58)(49,61,51,63)(50,64,52,62), (1,54)(2,55)(3,56)(4,53)(5,41)(6,42)(7,43)(8,44)(9,33)(10,34)(11,35)(12,36)(13,25)(14,26)(15,27)(16,28)(17,58)(18,59)(19,60)(20,57)(21,63)(22,64)(23,61)(24,62)(29,49)(30,50)(31,51)(32,52)(37,47)(38,48)(39,45)(40,46), (1,23,10,57)(2,24,11,58)(3,21,12,59)(4,22,9,60)(5,48,27,32)(6,45,28,29)(7,46,25,30)(8,47,26,31)(13,50,43,40)(14,51,44,37)(15,52,41,38)(16,49,42,39)(17,55,62,35)(18,56,63,36)(19,53,64,33)(20,54,61,34) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,9),(5,27),(6,28),(7,25),(8,26),(13,43),(14,44),(15,41),(16,42),(17,62),(18,63),(19,64),(20,61),(21,59),(22,60),(23,57),(24,58),(29,45),(30,46),(31,47),(32,48),(33,53),(34,54),(35,55),(36,56),(37,51),(38,52),(39,49),(40,50)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(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)], [(1,32,3,30),(2,45,4,47),(5,21,7,23),(6,60,8,58),(9,31,11,29),(10,48,12,46),(13,61,15,63),(14,17,16,19),(18,43,20,41),(22,26,24,28),(25,57,27,59),(33,37,35,39),(34,52,36,50),(38,56,40,54),(42,64,44,62),(49,53,51,55)], [(1,26,3,28),(2,25,4,27),(5,11,7,9),(6,10,8,12),(13,53,15,55),(14,56,16,54),(17,40,19,38),(18,39,20,37),(21,29,23,31),(22,32,24,30),(33,41,35,43),(34,44,36,42),(45,57,47,59),(46,60,48,58),(49,61,51,63),(50,64,52,62)], [(1,54),(2,55),(3,56),(4,53),(5,41),(6,42),(7,43),(8,44),(9,33),(10,34),(11,35),(12,36),(13,25),(14,26),(15,27),(16,28),(17,58),(18,59),(19,60),(20,57),(21,63),(22,64),(23,61),(24,62),(29,49),(30,50),(31,51),(32,52),(37,47),(38,48),(39,45),(40,46)], [(1,23,10,57),(2,24,11,58),(3,21,12,59),(4,22,9,60),(5,48,27,32),(6,45,28,29),(7,46,25,30),(8,47,26,31),(13,50,43,40),(14,51,44,37),(15,52,41,38),(16,49,42,39),(17,55,62,35),(18,56,63,36),(19,53,64,33),(20,54,61,34)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4Z | 4AA | ··· | 4AL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | C2.C25 |
| kernel | C22.69C25 | C2×C4×Q8 | C4×C4○D4 | C23.36C23 | C22.26C24 | Q8⋊5D4 | Q8⋊6D4 | C22.47C24 | C22.50C24 | Q8⋊3Q8 | C22.53C24 | C2×C4 | Q8 | C2 |
| # reps | 1 | 1 | 5 | 6 | 3 | 2 | 1 | 6 | 3 | 1 | 3 | 8 | 8 | 2 |
Matrix representation of C22.69C25 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,3,0,0,3,0],[0,2,0,0,3,0,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1] >;
C22.69C25 in GAP, Magma, Sage, TeX
C_2^2._{69}C_2^5 % in TeX
G:=Group("C2^2.69C2^5"); // GroupNames label
G:=SmallGroup(128,2212);
// by ID
G=gap.SmallGroup(128,2212);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,456,1430,352,570,102]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=f^2=1,c^2=d^2=e^2=b,g^2=a,a*b=b*a,d*c*d^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,c*g=g*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations