p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.552C24, C24.376C23, C22.2432- (1+4), C22.3272+ (1+4), C23.72(C4○D4), (C2×C42).84C22, (C22×C4).162C23, (C23×C4).145C22, C23.8Q8.43C2, C23.Q8.22C2, C23.11D4.30C2, C23.81C23⋊68C2, C23.83C23⋊69C2, C2.50(C22.32C24), C23.63C23⋊119C2, C23.65C23⋊108C2, C2.C42.269C22, C2.59(C22.36C24), C2.33(C22.35C24), C2.49(C22.33C24), C2.104(C23.36C23), (C4×C22⋊C4).73C2, (C2×C4).177(C4○D4), (C2×C4⋊C4).377C22, C22.424(C2×C4○D4), (C2×C22⋊C4).473C22, SmallGroup(128,1384)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×16], C22 [×7], C22 [×10], C2×C4 [×4], C2×C4 [×44], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×14], C22×C4 [×14], C22×C4 [×4], C24, C2.C42 [×16], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×10], C23×C4, C4×C22⋊C4, C23.8Q8 [×2], C23.63C23 [×3], C23.65C23, C23.Q8, C23.11D4 [×3], C23.81C23, C23.83C23 [×3], C23.552C24
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ (1+4) [×2], 2- (1+4) [×2], C23.36C23, C22.32C24, C22.33C24 [×3], C22.35C24, C22.36C24, C23.552C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=a, g2=b, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(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)
(2 10)(4 12)(5 40)(6 8)(7 38)(14 42)(16 44)(17 19)(18 48)(20 46)(22 50)(24 52)(26 54)(28 56)(29 31)(30 60)(32 58)(33 35)(34 62)(36 64)(37 39)(45 47)(57 59)(61 63)
(1 17 9 45)(2 32 10 60)(3 19 11 47)(4 30 12 58)(5 54 38 26)(6 41 39 13)(7 56 40 28)(8 43 37 15)(14 64 42 34)(16 62 44 36)(18 24 46 52)(20 22 48 50)(21 57 49 29)(23 59 51 31)(25 61 53 35)(27 63 55 33)
(1 55 51 41)(2 56 52 42)(3 53 49 43)(4 54 50 44)(5 20 62 30)(6 17 63 31)(7 18 64 32)(8 19 61 29)(9 27 23 13)(10 28 24 14)(11 25 21 15)(12 26 22 16)(33 59 39 45)(34 60 40 46)(35 57 37 47)(36 58 38 48)
G:=sub<Sym(64)| (1,9)(2,10)(3,11)(4,12)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,62)(6,63)(7,64)(8,61)(9,23)(10,24)(11,21)(12,22)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,39)(34,40)(35,37)(36,38)(41,55)(42,56)(43,53)(44,54)(45,59)(46,60)(47,57)(48,58), (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), (2,10)(4,12)(5,40)(6,8)(7,38)(14,42)(16,44)(17,19)(18,48)(20,46)(22,50)(24,52)(26,54)(28,56)(29,31)(30,60)(32,58)(33,35)(34,62)(36,64)(37,39)(45,47)(57,59)(61,63), (1,17,9,45)(2,32,10,60)(3,19,11,47)(4,30,12,58)(5,54,38,26)(6,41,39,13)(7,56,40,28)(8,43,37,15)(14,64,42,34)(16,62,44,36)(18,24,46,52)(20,22,48,50)(21,57,49,29)(23,59,51,31)(25,61,53,35)(27,63,55,33), (1,55,51,41)(2,56,52,42)(3,53,49,43)(4,54,50,44)(5,20,62,30)(6,17,63,31)(7,18,64,32)(8,19,61,29)(9,27,23,13)(10,28,24,14)(11,25,21,15)(12,26,22,16)(33,59,39,45)(34,60,40,46)(35,57,37,47)(36,58,38,48)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,62)(6,63)(7,64)(8,61)(9,23)(10,24)(11,21)(12,22)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,39)(34,40)(35,37)(36,38)(41,55)(42,56)(43,53)(44,54)(45,59)(46,60)(47,57)(48,58), (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), (2,10)(4,12)(5,40)(6,8)(7,38)(14,42)(16,44)(17,19)(18,48)(20,46)(22,50)(24,52)(26,54)(28,56)(29,31)(30,60)(32,58)(33,35)(34,62)(36,64)(37,39)(45,47)(57,59)(61,63), (1,17,9,45)(2,32,10,60)(3,19,11,47)(4,30,12,58)(5,54,38,26)(6,41,39,13)(7,56,40,28)(8,43,37,15)(14,64,42,34)(16,62,44,36)(18,24,46,52)(20,22,48,50)(21,57,49,29)(23,59,51,31)(25,61,53,35)(27,63,55,33), (1,55,51,41)(2,56,52,42)(3,53,49,43)(4,54,50,44)(5,20,62,30)(6,17,63,31)(7,18,64,32)(8,19,61,29)(9,27,23,13)(10,28,24,14)(11,25,21,15)(12,26,22,16)(33,59,39,45)(34,60,40,46)(35,57,37,47)(36,58,38,48) );
G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,38),(6,39),(7,40),(8,37),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,63),(34,64),(35,61),(36,62)], [(1,51),(2,52),(3,49),(4,50),(5,62),(6,63),(7,64),(8,61),(9,23),(10,24),(11,21),(12,22),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30),(33,39),(34,40),(35,37),(36,38),(41,55),(42,56),(43,53),(44,54),(45,59),(46,60),(47,57),(48,58)], [(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)], [(2,10),(4,12),(5,40),(6,8),(7,38),(14,42),(16,44),(17,19),(18,48),(20,46),(22,50),(24,52),(26,54),(28,56),(29,31),(30,60),(32,58),(33,35),(34,62),(36,64),(37,39),(45,47),(57,59),(61,63)], [(1,17,9,45),(2,32,10,60),(3,19,11,47),(4,30,12,58),(5,54,38,26),(6,41,39,13),(7,56,40,28),(8,43,37,15),(14,64,42,34),(16,62,44,36),(18,24,46,52),(20,22,48,50),(21,57,49,29),(23,59,51,31),(25,61,53,35),(27,63,55,33)], [(1,55,51,41),(2,56,52,42),(3,53,49,43),(4,54,50,44),(5,20,62,30),(6,17,63,31),(7,18,64,32),(8,19,61,29),(9,27,23,13),(10,28,24,14),(11,25,21,15),(12,26,22,16),(33,59,39,45),(34,60,40,46),(35,57,37,47),(36,58,38,48)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 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 |
| 1 | 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 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.552C24 | C4×C22⋊C4 | C23.8Q8 | C23.63C23 | C23.65C23 | C23.Q8 | C23.11D4 | C23.81C23 | C23.83C23 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 3 | 8 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{552}C_2^4 % in TeX
G:=Group("C2^3.552C2^4"); // GroupNames label
G:=SmallGroup(128,1384);
// by ID
G=gap.SmallGroup(128,1384);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c,f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*g=g*e>;
// generators/relations