p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.380C23, C23.567C24, C22.3412+ (1+4), C22.2562- (1+4), C23.70(C2×Q8), (C22×C4).62Q8, (C22×C4).172C23, (C23×C4).440C22, (C2×C42).631C22, C2.12(C23⋊2Q8), C23.Q8.26C2, C23.7Q8.63C2, C23.4Q8.18C2, C22.141(C22×Q8), C23.34D4.25C2, C23.81C23⋊74C2, C23.65C23⋊112C2, C23.63C23⋊123C2, C2.C42.281C22, C2.56(C22.33C24), C2.37(C22.34C24), C2.28(C23.41C23), C2.44(C23.37C23), (C2×C4).170(C2×Q8), (C4×C22⋊C4).75C2, (C2×C4).187(C4○D4), (C2×C4⋊C4).388C22, C22.434(C2×C4○D4), (C2×C22⋊C4).523C22, SmallGroup(128,1399)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 204 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×18], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×8], C2×C4 [×42], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×18], C22×C4 [×2], C22×C4 [×16], C22×C4 [×2], C24, C2.C42 [×4], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×14], C23×C4, C4×C22⋊C4, C23.7Q8, C23.34D4, C23.63C23 [×2], C23.65C23 [×2], C23.Q8 [×2], C23.81C23 [×4], C23.4Q8 [×2], C24.380C23
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C23.37C23, C22.33C24 [×2], C22.34C24 [×2], C23⋊2Q8, C23.41C23, C24.380C23
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=db=bd, g2=c, eae-1=ab=ba, faf-1=ac=ca, ad=da, ag=ga, bc=cb, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge >
(2 10)(4 12)(5 36)(6 63)(7 34)(8 61)(14 42)(16 44)(17 31)(18 60)(19 29)(20 58)(22 50)(24 52)(26 54)(28 56)(30 48)(32 46)(33 39)(35 37)(38 62)(40 64)(45 59)(47 57)
(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)
(1 19 11 45)(2 18 12 48)(3 17 9 47)(4 20 10 46)(5 42 40 16)(6 41 37 15)(7 44 38 14)(8 43 39 13)(21 59 51 29)(22 58 52 32)(23 57 49 31)(24 60 50 30)(25 63 55 35)(26 62 56 34)(27 61 53 33)(28 64 54 36)
(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)| (2,10)(4,12)(5,36)(6,63)(7,34)(8,61)(14,42)(16,44)(17,31)(18,60)(19,29)(20,58)(22,50)(24,52)(26,54)(28,56)(30,48)(32,46)(33,39)(35,37)(38,62)(40,64)(45,59)(47,57), (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), (1,19,11,45)(2,18,12,48)(3,17,9,47)(4,20,10,46)(5,42,40,16)(6,41,37,15)(7,44,38,14)(8,43,39,13)(21,59,51,29)(22,58,52,32)(23,57,49,31)(24,60,50,30)(25,63,55,35)(26,62,56,34)(27,61,53,33)(28,64,54,36), (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( (2,10)(4,12)(5,36)(6,63)(7,34)(8,61)(14,42)(16,44)(17,31)(18,60)(19,29)(20,58)(22,50)(24,52)(26,54)(28,56)(30,48)(32,46)(33,39)(35,37)(38,62)(40,64)(45,59)(47,57), (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), (1,19,11,45)(2,18,12,48)(3,17,9,47)(4,20,10,46)(5,42,40,16)(6,41,37,15)(7,44,38,14)(8,43,39,13)(21,59,51,29)(22,58,52,32)(23,57,49,31)(24,60,50,30)(25,63,55,35)(26,62,56,34)(27,61,53,33)(28,64,54,36), (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([(2,10),(4,12),(5,36),(6,63),(7,34),(8,61),(14,42),(16,44),(17,31),(18,60),(19,29),(20,58),(22,50),(24,52),(26,54),(28,56),(30,48),(32,46),(33,39),(35,37),(38,62),(40,64),(45,59),(47,57)], [(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)], [(1,19,11,45),(2,18,12,48),(3,17,9,47),(4,20,10,46),(5,42,40,16),(6,41,37,15),(7,44,38,14),(8,43,39,13),(21,59,51,29),(22,58,52,32),(23,57,49,31),(24,60,50,30),(25,63,55,35),(26,62,56,34),(27,61,53,33),(28,64,54,36)], [(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 | 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 | 4 | 3 | 0 | 4 |
| 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 |
| 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 |
| 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 | 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 |
| 3 | 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 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 1 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 3 |
| 0 | 1 | 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 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 | 1 | 4 |
| 0 | 0 | 0 | 0 | 1 | 2 | 2 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 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 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 3 | 1 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,1,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,2,0,1,0,0,0,0,1,1,0,1,0,0,0,0,0,2,0,3],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,4,2,1,0,0,0,0,2,1,4,2,0,0,0,0,0,0,1,2,0,0,0,0,0,0,4,4],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,4,0,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,1] >;
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 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C24.380C23 | C4×C22⋊C4 | C23.7Q8 | C23.34D4 | C23.63C23 | C23.65C23 | C23.Q8 | C23.81C23 | C23.4Q8 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 3 | 1 |
In GAP, Magma, Sage, TeX
C_2^4._{380}C_2^3 % in TeX
G:=Group("C2^4.380C2^3"); // GroupNames label
G:=SmallGroup(128,1399);
// by ID
G=gap.SmallGroup(128,1399);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,436,185,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=d*b=b*d,g^2=c,e*a*e^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
// generators/relations