p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.7M4(2), C4⋊C4.1C8, (C2×C8).293D4, C22⋊C16.2C2, C2.5(D4.C8), C4.40(C23⋊C4), C42⋊C2.1C4, (C2×M4(2)).7C4, (C22×C8).4C22, C4.21(C4.10D4), C22.49(C22⋊C8), C42.6C22.8C2, C2.4(C22.M4(2)), (C2×C4).11(C2×C8), (C22×C4).165(C2×C4), (C2×C4).379(C22⋊C4), SmallGroup(128,55)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.7M4(2)
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, eae-1=ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abcd5 >
(2 19)(4 21)(6 23)(8 25)(10 27)(12 29)(14 31)(16 17)(33 41)(34 54)(35 43)(36 56)(37 45)(38 58)(39 47)(40 60)(42 62)(44 64)(46 50)(48 52)(49 57)(51 59)(53 61)(55 63)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 49)(46 50)(47 51)(48 52)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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 39 26 59)(2 64 27 44)(3 61 28 41)(4 46 29 50)(5 43 30 63)(6 52 31 48)(7 49 32 45)(8 34 17 54)(9 47 18 51)(10 56 19 36)(11 53 20 33)(12 38 21 58)(13 35 22 55)(14 60 23 40)(15 57 24 37)(16 42 25 62)
G:=sub<Sym(64)| (2,19)(4,21)(6,23)(8,25)(10,27)(12,29)(14,31)(16,17)(33,41)(34,54)(35,43)(36,56)(37,45)(38,58)(39,47)(40,60)(42,62)(44,64)(46,50)(48,52)(49,57)(51,59)(53,61)(55,63), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,39,26,59)(2,64,27,44)(3,61,28,41)(4,46,29,50)(5,43,30,63)(6,52,31,48)(7,49,32,45)(8,34,17,54)(9,47,18,51)(10,56,19,36)(11,53,20,33)(12,38,21,58)(13,35,22,55)(14,60,23,40)(15,57,24,37)(16,42,25,62)>;
G:=Group( (2,19)(4,21)(6,23)(8,25)(10,27)(12,29)(14,31)(16,17)(33,41)(34,54)(35,43)(36,56)(37,45)(38,58)(39,47)(40,60)(42,62)(44,64)(46,50)(48,52)(49,57)(51,59)(53,61)(55,63), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,39,26,59)(2,64,27,44)(3,61,28,41)(4,46,29,50)(5,43,30,63)(6,52,31,48)(7,49,32,45)(8,34,17,54)(9,47,18,51)(10,56,19,36)(11,53,20,33)(12,38,21,58)(13,35,22,55)(14,60,23,40)(15,57,24,37)(16,42,25,62) );
G=PermutationGroup([(2,19),(4,21),(6,23),(8,25),(10,27),(12,29),(14,31),(16,17),(33,41),(34,54),(35,43),(36,56),(37,45),(38,58),(39,47),(40,60),(42,62),(44,64),(46,50),(48,52),(49,57),(51,59),(53,61),(55,63)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,17),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,25),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,49),(46,50),(47,51),(48,52)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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)], [(1,39,26,59),(2,64,27,44),(3,61,28,41),(4,46,29,50),(5,43,30,63),(6,52,31,48),(7,49,32,45),(8,34,17,54),(9,47,18,51),(10,56,19,36),(11,53,20,33),(12,38,21,58),(13,35,22,55),(14,60,23,40),(15,57,24,37),(16,42,25,62)])
38 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | ··· | 8H | 8I | 8J | 16A | ··· | 16P |
order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 16 | ··· | 16 |
size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | - | |||||
image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | D4.C8 | C23⋊C4 | C4.10D4 |
kernel | C23.7M4(2) | C22⋊C16 | C42.6C22 | C42⋊C2 | C2×M4(2) | C4⋊C4 | C2×C8 | C23 | C2 | C4 | C4 |
# reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 16 | 1 | 1 |
Matrix representation of C23.7M4(2) ►in GL4(𝔽17) generated by
1 | 0 | 0 | 0 |
13 | 16 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 2 | 16 |
16 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
16 | 0 | 0 | 0 |
0 | 16 | 0 | 0 |
0 | 0 | 16 | 0 |
0 | 0 | 0 | 16 |
5 | 0 | 0 | 0 |
1 | 14 | 0 | 0 |
0 | 0 | 12 | 5 |
0 | 0 | 15 | 5 |
13 | 15 | 0 | 0 |
0 | 4 | 0 | 0 |
0 | 0 | 16 | 1 |
0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,13,0,0,0,16,0,0,0,0,1,2,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[5,1,0,0,0,14,0,0,0,0,12,15,0,0,5,5],[13,0,0,0,15,4,0,0,0,0,16,0,0,0,1,1] >;
C23.7M4(2) in GAP, Magma, Sage, TeX
C_2^3._7M_4(2)
% in TeX
G:=Group("C2^3.7M4(2)");
// GroupNames label
G:=SmallGroup(128,55);
// by ID
G=gap.SmallGroup(128,55);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,723,352,1242,136,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b*c*d^5>;
// generators/relations
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