p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.83C23, C22.91C25, C23.135C24, C4.402- 1+4, Q82⋊8C2, (C2×Q8)⋊16Q8, Q8○2(C22⋊Q8), Q8.15(C2×Q8), Q8⋊3Q8⋊15C2, (C2×C4).81C24, C4.54(C22×Q8), C2.16(Q8×C23), C4⋊C4.297C23, C4⋊Q8.341C22, C22.8(C22×Q8), (C4×Q8).219C22, (C2×Q8).488C23, C22⋊C4.101C23, (C2×C42).945C22, C22⋊Q8.241C22, C2.24(C2×2- 1+4), C42.C2.81C22, C2.26(C2.C25), (C22×C4).1209C23, (C22×Q8).499C22, C42⋊C2.226C22, C23.32C23.7C2, C23.37C23.45C2, C23.41C23.11C2, (C2×C4×Q8).57C2, (C2×Q8)○(C22⋊Q8), (C2×C4).109(C2×Q8), (C2×C4⋊C4).964C22, SmallGroup(128,2234)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.91C25
G = < a,b,c,d,e,f,g | a2=b2=g2=1, c2=e2=b, d2=f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 564 in 482 conjugacy classes, 430 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×Q8, C2×C4×Q8, C23.32C23, C23.37C23, C23.41C23, Q8⋊3Q8, Q82, C22.91C25
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2- 1+4, C25, Q8×C23, C2×2- 1+4, C2.C25, C22.91C25
►On 64 points
Generators in S64
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 59 51 47)(2 48 52 60)(3 57 49 45)(4 46 50 58)(5 26 36 14)(6 15 33 27)(7 28 34 16)(8 13 35 25)(9 29 21 17)(10 18 22 30)(11 31 23 19)(12 20 24 32)(37 41 61 53)(38 54 62 42)(39 43 63 55)(40 56 64 44)
(1 41 3 43)(2 44 4 42)(5 20 7 18)(6 19 8 17)(9 15 11 13)(10 14 12 16)(21 27 23 25)(22 26 24 28)(29 33 31 35)(30 36 32 34)(37 45 39 47)(38 48 40 46)(49 55 51 53)(50 54 52 56)(57 63 59 61)(58 62 60 64)
(1 23 51 11)(2 24 52 12)(3 21 49 9)(4 22 50 10)(5 62 36 38)(6 63 33 39)(7 64 34 40)(8 61 35 37)(13 41 25 53)(14 42 26 54)(15 43 27 55)(16 44 28 56)(17 57 29 45)(18 58 30 46)(19 59 31 47)(20 60 32 48)
(2 52)(4 50)(5 36)(7 34)(10 22)(12 24)(14 26)(16 28)(18 30)(20 32)(38 62)(40 64)(42 54)(44 56)(46 58)(48 60)
G:=sub<Sym(64)| (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,59,51,47)(2,48,52,60)(3,57,49,45)(4,46,50,58)(5,26,36,14)(6,15,33,27)(7,28,34,16)(8,13,35,25)(9,29,21,17)(10,18,22,30)(11,31,23,19)(12,20,24,32)(37,41,61,53)(38,54,62,42)(39,43,63,55)(40,56,64,44), (1,41,3,43)(2,44,4,42)(5,20,7,18)(6,19,8,17)(9,15,11,13)(10,14,12,16)(21,27,23,25)(22,26,24,28)(29,33,31,35)(30,36,32,34)(37,45,39,47)(38,48,40,46)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64), (1,23,51,11)(2,24,52,12)(3,21,49,9)(4,22,50,10)(5,62,36,38)(6,63,33,39)(7,64,34,40)(8,61,35,37)(13,41,25,53)(14,42,26,54)(15,43,27,55)(16,44,28,56)(17,57,29,45)(18,58,30,46)(19,59,31,47)(20,60,32,48), (2,52)(4,50)(5,36)(7,34)(10,22)(12,24)(14,26)(16,28)(18,30)(20,32)(38,62)(40,64)(42,54)(44,56)(46,58)(48,60)>;
G:=Group( (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,59,51,47)(2,48,52,60)(3,57,49,45)(4,46,50,58)(5,26,36,14)(6,15,33,27)(7,28,34,16)(8,13,35,25)(9,29,21,17)(10,18,22,30)(11,31,23,19)(12,20,24,32)(37,41,61,53)(38,54,62,42)(39,43,63,55)(40,56,64,44), (1,41,3,43)(2,44,4,42)(5,20,7,18)(6,19,8,17)(9,15,11,13)(10,14,12,16)(21,27,23,25)(22,26,24,28)(29,33,31,35)(30,36,32,34)(37,45,39,47)(38,48,40,46)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64), (1,23,51,11)(2,24,52,12)(3,21,49,9)(4,22,50,10)(5,62,36,38)(6,63,33,39)(7,64,34,40)(8,61,35,37)(13,41,25,53)(14,42,26,54)(15,43,27,55)(16,44,28,56)(17,57,29,45)(18,58,30,46)(19,59,31,47)(20,60,32,48), (2,52)(4,50)(5,36)(7,34)(10,22)(12,24)(14,26)(16,28)(18,30)(20,32)(38,62)(40,64)(42,54)(44,56)(46,58)(48,60) );
G=PermutationGroup([[(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(37,61),(38,62),(39,63),(40,64),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(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)], [(1,59,51,47),(2,48,52,60),(3,57,49,45),(4,46,50,58),(5,26,36,14),(6,15,33,27),(7,28,34,16),(8,13,35,25),(9,29,21,17),(10,18,22,30),(11,31,23,19),(12,20,24,32),(37,41,61,53),(38,54,62,42),(39,43,63,55),(40,56,64,44)], [(1,41,3,43),(2,44,4,42),(5,20,7,18),(6,19,8,17),(9,15,11,13),(10,14,12,16),(21,27,23,25),(22,26,24,28),(29,33,31,35),(30,36,32,34),(37,45,39,47),(38,48,40,46),(49,55,51,53),(50,54,52,56),(57,63,59,61),(58,62,60,64)], [(1,23,51,11),(2,24,52,12),(3,21,49,9),(4,22,50,10),(5,62,36,38),(6,63,33,39),(7,64,34,40),(8,61,35,37),(13,41,25,53),(14,42,26,54),(15,43,27,55),(16,44,28,56),(17,57,29,45),(18,58,30,46),(19,59,31,47),(20,60,32,48)], [(2,52),(4,50),(5,36),(7,34),(10,22),(12,24),(14,26),(16,28),(18,30),(20,32),(38,62),(40,64),(42,54),(44,56),(46,58),(48,60)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4P | 4Q | ··· | 4AL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | 2- 1+4 | C2.C25 |
| kernel | C22.91C25 | C2×C4×Q8 | C23.32C23 | C23.37C23 | C23.41C23 | Q8⋊3Q8 | Q82 | C2×Q8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 4 | 8 | 2 | 2 |
Matrix representation of C22.91C25 ►in GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
C22.91C25 in GAP, Magma, Sage, TeX
C_2^2._{91}C_2^5 % in TeX
G:=Group("C2^2.91C2^5"); // GroupNames label
G:=SmallGroup(128,2234);
// by ID
G=gap.SmallGroup(128,2234);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,1430,352,570,136,1684]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=g^2=1,c^2=e^2=b,d^2=f^2=a,a*b=b*a,d*c*d^-1=g*c*g=a*c=c*a,f*d*f^-1=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,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations