p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.279C23, C22⋊C4○Q16, C4⋊C4.402D4, (C2×Q16)⋊18C4, (C4×Q16)⋊38C2, C4.103(C4×D4), C2.4(Q8○D8), Q16⋊C4⋊3C2, C8.22(C22×C4), C4.27(C23×C4), Q16.14(C2×C4), C22.70(C4×D4), Q8.9(C22×C4), C4⋊C4.367C23, (C2×C4).207C24, (C4×C8).221C22, (C2×C8).418C23, C22⋊C4.189D4, C23.439(C2×D4), (C4×Q8).54C22, C8○2M4(2).8C2, (C22×Q16).16C2, (C2×Q8).348C23, C2.D8.235C22, C8⋊C4.114C22, C4.Q8.128C22, (C22×C4).928C23, (C22×C8).250C22, (C2×Q16).154C22, C22.151(C22×D4), C42⋊C2.84C22, Q8⋊C4.198C22, C23.38D4.10C2, C23.25D4.17C2, (C22×Q8).260C22, (C2×M4(2)).354C22, C23.32C23.4C2, C2.67(C2×C4×D4), (C2×C8).97(C2×C4), C4.15(C2×C4○D4), (C2×C4).914(C2×D4), (C2×Q8).115(C2×C4), (C2×C4).266(C4○D4), (C2×C4).266(C22×C4), SmallGroup(128,1682)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.279C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=b2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=b2c, de=ed >
Subgroups: 324 in 228 conjugacy classes, 140 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C42⋊C2, C4×Q8, C4×Q8, C22×C8, C2×M4(2), C2×Q16, C22×Q8, C8○2M4(2), C23.38D4, C23.25D4, C4×Q16, Q16⋊C4, C23.32C23, C22×Q16, C42.279C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, Q8○D8, C42.279C23
►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 13 7 19)(2 14 8 20)(3 15 5 17)(4 16 6 18)(9 61 60 53)(10 62 57 54)(11 63 58 55)(12 64 59 56)(21 28 29 33)(22 25 30 34)(23 26 31 35)(24 27 32 36)(37 44 45 49)(38 41 46 50)(39 42 47 51)(40 43 48 52)
(1 37 5 47)(2 38 6 48)(3 39 7 45)(4 40 8 46)(9 33 58 26)(10 34 59 27)(11 35 60 28)(12 36 57 25)(13 49 17 42)(14 50 18 43)(15 51 19 44)(16 52 20 41)(21 63 31 53)(22 64 32 54)(23 61 29 55)(24 62 30 56)
(1 23 7 31)(2 24 8 32)(3 21 5 29)(4 22 6 30)(9 45 60 37)(10 46 57 38)(11 47 58 39)(12 48 59 40)(13 35 19 26)(14 36 20 27)(15 33 17 28)(16 34 18 25)(41 54 50 62)(42 55 51 63)(43 56 52 64)(44 53 49 61)
(2 8)(4 6)(9 60)(11 58)(14 20)(16 18)(22 30)(24 32)(25 34)(27 36)(37 45)(39 47)(42 51)(44 49)(53 61)(55 63)
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), (1,13,7,19)(2,14,8,20)(3,15,5,17)(4,16,6,18)(9,61,60,53)(10,62,57,54)(11,63,58,55)(12,64,59,56)(21,28,29,33)(22,25,30,34)(23,26,31,35)(24,27,32,36)(37,44,45,49)(38,41,46,50)(39,42,47,51)(40,43,48,52), (1,37,5,47)(2,38,6,48)(3,39,7,45)(4,40,8,46)(9,33,58,26)(10,34,59,27)(11,35,60,28)(12,36,57,25)(13,49,17,42)(14,50,18,43)(15,51,19,44)(16,52,20,41)(21,63,31,53)(22,64,32,54)(23,61,29,55)(24,62,30,56), (1,23,7,31)(2,24,8,32)(3,21,5,29)(4,22,6,30)(9,45,60,37)(10,46,57,38)(11,47,58,39)(12,48,59,40)(13,35,19,26)(14,36,20,27)(15,33,17,28)(16,34,18,25)(41,54,50,62)(42,55,51,63)(43,56,52,64)(44,53,49,61), (2,8)(4,6)(9,60)(11,58)(14,20)(16,18)(22,30)(24,32)(25,34)(27,36)(37,45)(39,47)(42,51)(44,49)(53,61)(55,63)>;
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), (1,13,7,19)(2,14,8,20)(3,15,5,17)(4,16,6,18)(9,61,60,53)(10,62,57,54)(11,63,58,55)(12,64,59,56)(21,28,29,33)(22,25,30,34)(23,26,31,35)(24,27,32,36)(37,44,45,49)(38,41,46,50)(39,42,47,51)(40,43,48,52), (1,37,5,47)(2,38,6,48)(3,39,7,45)(4,40,8,46)(9,33,58,26)(10,34,59,27)(11,35,60,28)(12,36,57,25)(13,49,17,42)(14,50,18,43)(15,51,19,44)(16,52,20,41)(21,63,31,53)(22,64,32,54)(23,61,29,55)(24,62,30,56), (1,23,7,31)(2,24,8,32)(3,21,5,29)(4,22,6,30)(9,45,60,37)(10,46,57,38)(11,47,58,39)(12,48,59,40)(13,35,19,26)(14,36,20,27)(15,33,17,28)(16,34,18,25)(41,54,50,62)(42,55,51,63)(43,56,52,64)(44,53,49,61), (2,8)(4,6)(9,60)(11,58)(14,20)(16,18)(22,30)(24,32)(25,34)(27,36)(37,45)(39,47)(42,51)(44,49)(53,61)(55,63) );
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)], [(1,13,7,19),(2,14,8,20),(3,15,5,17),(4,16,6,18),(9,61,60,53),(10,62,57,54),(11,63,58,55),(12,64,59,56),(21,28,29,33),(22,25,30,34),(23,26,31,35),(24,27,32,36),(37,44,45,49),(38,41,46,50),(39,42,47,51),(40,43,48,52)], [(1,37,5,47),(2,38,6,48),(3,39,7,45),(4,40,8,46),(9,33,58,26),(10,34,59,27),(11,35,60,28),(12,36,57,25),(13,49,17,42),(14,50,18,43),(15,51,19,44),(16,52,20,41),(21,63,31,53),(22,64,32,54),(23,61,29,55),(24,62,30,56)], [(1,23,7,31),(2,24,8,32),(3,21,5,29),(4,22,6,30),(9,45,60,37),(10,46,57,38),(11,47,58,39),(12,48,59,40),(13,35,19,26),(14,36,20,27),(15,33,17,28),(16,34,18,25),(41,54,50,62),(42,55,51,63),(43,56,52,64),(44,53,49,61)], [(2,8),(4,6),(9,60),(11,58),(14,20),(16,18),(22,30),(24,32),(25,34),(27,36),(37,45),(39,47),(42,51),(44,49),(53,61),(55,63)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4AB | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
44 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 | C4 | D4 | D4 | C4○D4 | Q8○D8 |
| kernel | C42.279C23 | C8○2M4(2) | C23.38D4 | C23.25D4 | C4×Q16 | Q16⋊C4 | C23.32C23 | C22×Q16 | C2×Q16 | C22⋊C4 | C4⋊C4 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C42.279C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,5,12,0,0,0,0,12,12,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
C42.279C23 in GAP, Magma, Sage, TeX
C_4^2._{279}C_2^3 % in TeX
G:=Group("C4^2.279C2^3"); // GroupNames label
G:=SmallGroup(128,1682);
// by ID
G=gap.SmallGroup(128,1682);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,520,2019,521,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=b^2*c,d*e=e*d>;
// generators/relations