p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.37C23, C42.36C22, C22.27C24, (C2×C4)⋊5Q8, C4○2(C4⋊Q8), C4⋊Q8⋊20C2, (C4×Q8)⋊8C2, C4.25(C2×Q8), C4○3(C22⋊Q8), C4.19(C4○D4), C22.3(C2×Q8), C2.6(C22×Q8), C4⋊C4.73C22, (C2×C4).15C23, (C2×C42).20C2, C4○2(C42.C2), C42.C2⋊14C2, C22⋊Q8.10C2, (C2×Q8).57C22, C42⋊C2.11C2, C22⋊C4.15C22, (C22×C4).126C22, (C2×C4)○(C4⋊Q8), C2.14(C2×C4○D4), SmallGroup(64,214)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.37C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e2=c, f2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ede-1=cd=dc, ce=ec, cf=fc, df=fd, ef=fe >
Subgroups: 137 in 111 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23.37C23
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, C23.37C23
Character table of C23.37C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 2i | -2 | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 2i | 2 | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | 2i | -2i | -2i | -2 | 2i | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | -2i | 2i | 2i | -2 | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | -2i | -2 | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | -2i | 2i | 2i | 2 | 2i | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | 2i | -2i | -2i | 2 | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | -2i | 2 | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 32)
(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)
(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)
(1 21 3 23)(2 24 4 22)(5 12 7 10)(6 11 8 9)(13 17 15 19)(14 20 16 18)(25 31 27 29)(26 30 28 32)
(1 15 27 11)(2 16 28 12)(3 13 25 9)(4 14 26 10)(5 22 20 30)(6 23 17 31)(7 24 18 32)(8 21 19 29)
G:=sub<Sym(32)| (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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), (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), (1,21,3,23)(2,24,4,22)(5,12,7,10)(6,11,8,9)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32), (1,15,27,11)(2,16,28,12)(3,13,25,9)(4,14,26,10)(5,22,20,30)(6,23,17,31)(7,24,18,32)(8,21,19,29)>;
G:=Group( (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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), (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), (1,21,3,23)(2,24,4,22)(5,12,7,10)(6,11,8,9)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32), (1,15,27,11)(2,16,28,12)(3,13,25,9)(4,14,26,10)(5,22,20,30)(6,23,17,31)(7,24,18,32)(8,21,19,29) );
G=PermutationGroup([[(2,28),(4,26),(5,20),(7,18),(10,14),(12,16),(22,30),(24,32)], [(1,27),(2,28),(3,25),(4,26),(5,20),(6,17),(7,18),(8,19),(9,13),(10,14),(11,15),(12,16),(21,29),(22,30),(23,31),(24,32)], [(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)], [(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)], [(1,21,3,23),(2,24,4,22),(5,12,7,10),(6,11,8,9),(13,17,15,19),(14,20,16,18),(25,31,27,29),(26,30,28,32)], [(1,15,27,11),(2,16,28,12),(3,13,25,9),(4,14,26,10),(5,22,20,30),(6,23,17,31),(7,24,18,32),(8,21,19,29)]])
C23.37C23 is a maximal subgroup of
C22.33C25 C22.44C25 C22.47C25 C22.50C25 C22.64C25 Q8×C4○D4 C22.71C25 C22.84C25 C22.90C25 C22.91C25 C22.92C25 C22.93C25 C22.95C25 C22.96C25 C22.97C25 C22.98C25 C22.99C25 C22.100C25 C22.104C25 C22.107C25 C23.144C24 C23.146C24 C22.120C25 C22.136C25 C22.137C25 C22.139C25 C22.143C25 C22.144C25 C22.145C25 C22.146C25 C22.150C25 C22.152C25 C22.153C25 C22.154C25
C42.D2p: C42.46D4 C42.401D4 C42.316D4 C42.54D4 C42.404D4 C42.56D4 C42.60D4 C42.62D4 ...
(C2×C4p).C23: C42.286C23 C42.287C23 M4(2)⋊9Q8 C42.696C23 C42.304C23 C42.305C23 (Q8×Dic3)⋊C2 (Q8×Dic5)⋊C2 ...
C23.37C23 is a maximal quotient of
C23.167C24 C42⋊14Q8 C23.178C24 C4×C42.C2 C42.34Q8 C23.323C24 C24.567C23 C23.346C24 C23.397C24 C23.407C24 C23.411C24 C23.420C24 C23.422C24 C23.449C24 C24.584C23 C42.36Q8 C24.338C23 C23.485C24 C23.486C24 C24.345C23 C23.488C24 C24.346C23 C23.490C24 C42⋊8Q8 C42.38Q8 C24.355C23 C23.508C24 C42⋊9Q8 C24.379C23 C42⋊11Q8 C23.567C24 C24.599C23 C42⋊15Q8 C43.18C2
C42.D2p: C4×C22⋊Q8 C4×C4⋊Q8 C42.162D4 C42⋊5Q8 C42.439D4 C42.440D4 C43.15C2 C42⋊18Q8 ...
C4⋊C4.D2p: C23.329C24 C24.267C23 C24.268C23 C23.351C24 C23.362C24 C24.285C23 C23.392C24 C42⋊6Q8 ...
Matrix representation of C23.37C23 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,4,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,3],[3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2] >;
C23.37C23 in GAP, Magma, Sage, TeX
C_2^3._{37}C_2^3 % in TeX
G:=Group("C2^3.37C2^3"); // GroupNames label
G:=SmallGroup(64,214);
// by ID
G=gap.SmallGroup(64,214);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,650,158,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=e^2=c,f^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*d*e^-1=c*d=d*c,c*e=e*c,c*f=f*c,d*f=f*d,e*f=f*e>;
// generators/relations
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