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Elements of the group are displayed as permutations of degree 24.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_2^5.\POPlus(4,3)$ | 1A | $1$ | $1$ | $C_2^5.\POPlus(4,3)$ | 1A | 1A | $()$ |
| $C_2^5.\POPlus(4,3)$ | 2A | $2$ | $1$ | $C_2^5.\POPlus(4,3)$ | 1A | 2A | $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2B | $2$ | $3$ | $C_2^2.C_2^6.D_6.C_2$ | 1A | 2B | $(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2C | $2$ | $3$ | $C_2^2.C_2^6.D_6.C_2$ | 1A | 2C | $(17,18)(19,20)(21,22)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2D | $2$ | $12$ | $C_2\times C_4^3:D_6$ | 1A | 2D | $(5,6)(7,8)(13,14)(15,16)(21,22)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2E | $2$ | $12$ | $C_2\times C_4^3:D_6$ | 1A | 2E | $(1,2)(5,6)(11,12)(15,16)(19,20)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2F | $2$ | $24$ | $C_2^3.\GL(2,\mathbb{Z}/4)$ | 1A | 2F | $(1,8)(2,7)(3,6)(4,5)(9,13)(10,14)(11,15)(12,16)(17,19)(18,20)(21,23)(22,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2G | $2$ | $24$ | $C_2^3.\GL(2,\mathbb{Z}/4)$ | 1A | 2G | $(1,6)(2,5)(3,8)(4,7)(9,12)(10,11)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)$ |
| $C_2^5.\POPlus(4,3)$ | 2H | $2$ | $24$ | $C_2^5.S_4$ | 1A | 2H | $(1,18)(2,17)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(11,12)(13,14)(15,16)$ |
| $C_2^5.\POPlus(4,3)$ | 2I | $2$ | $24$ | $C_2^5.S_4$ | 1A | 2I | $(1,9)(2,10)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)(17,18)$ |
| $C_2^5.\POPlus(4,3)$ | 2J | $2$ | $72$ | $C_4^2:C_2^4$ | 1A | 2J | $(7,8)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,24)(16,23)$ |
| $C_2^5.\POPlus(4,3)$ | 2K | $2$ | $72$ | $C_4^2:C_2^4$ | 1A | 2K | $(5,6)(7,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,24)(20,23)$ |
| $C_2^5.\POPlus(4,3)$ | 2L | $2$ | $72$ | $C_4^2:C_2^4$ | 1A | 2L | $(5,6)(7,8)(9,13)(10,14)(11,15)(12,16)(17,22)(18,21)(19,23)(20,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2M | $2$ | $72$ | $C_4^2:C_2^4$ | 1A | 2M | $(3,4)(5,6)(7,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,23)(15,21)(16,22)$ |
| $C_2^5.\POPlus(4,3)$ | 2N | $2$ | $96$ | $C_2^3\times S_4$ | 1A | 2N | $(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(17,19)(18,20)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2O | $2$ | $96$ | $C_2^3\times S_4$ | 1A | 2O | $(1,2)(3,7)(4,8)(9,14)(10,13)(11,12)(17,22)(18,21)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 2P | $2$ | $144$ | not computed | 1A | 2P | $(1,12)(2,11)(3,13)(4,14)(5,15)(6,16)(7,10)(8,9)(17,22)(18,21)$ |
| $C_2^5.\POPlus(4,3)$ | 2Q | $2$ | $144$ | not computed | 1A | 2Q | $(1,2)(3,5)(4,6)(7,8)(9,24)(10,23)(11,20)(12,19)(13,21)(14,22)(15,17)(16,18)$ |
| $C_2^5.\POPlus(4,3)$ | 3A | $3$ | $128$ | $C_6\times S_4$ | 3A | 1A | $(3,7,6)(4,8,5)(11,15,14)(12,16,13)(19,23,22)(20,24,21)$ |
| $C_2^5.\POPlus(4,3)$ | 3B | $3$ | $128$ | $C_6\times S_4$ | 3B | 1A | $(1,17,10)(2,18,9)(3,19,12)(4,20,11)(5,21,14)(6,22,13)(7,23,16)(8,24,15)$ |
| $C_2^5.\POPlus(4,3)$ | 3C | $3$ | $256$ | not computed | 3C | 1A | $(1,17,9)(2,18,10)(3,21,16)(4,22,15)(5,24,11)(6,23,12)(7,20,14)(8,19,13)$ |
| $C_2^5.\POPlus(4,3)$ | 4A | $4$ | $18$ | $C_2^3.C_2^4.C_2^3$ | 2B | 4A | $(9,11,10,12)(13,16,14,15)(17,19,18,20)(21,23,22,24)$ |
| $C_2^5.\POPlus(4,3)$ | 4B | $4$ | $18$ | $C_2^2.C_2^5.C_2^3$ | 2B | 4B | $(1,2)(3,4)(5,6)(7,8)(9,11,10,12)(13,16,14,15)(17,19,18,20)(21,23,22,24)$ |
| $C_2^5.\POPlus(4,3)$ | 4C | $4$ | $18$ | $C_2^3.C_2^4.C_2^3$ | 2B | 4C | $(1,2)(3,4)(5,6)(7,8)(9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23)$ |
| $C_2^5.\POPlus(4,3)$ | 4D | $4$ | $18$ | $C_2^2.C_2^5.C_2^3$ | 2B | 4D | $(1,7,2,8)(3,5,4,6)(17,24,18,23)(19,21,20,22)$ |
| $C_2^5.\POPlus(4,3)$ | 4E | $4$ | $24$ | $C_2\times Q_8:C_2^2.D_6$ | 2D | 4E | $(1,4,2,3)(9,11,10,12)(21,24,22,23)$ |
| $C_2^5.\POPlus(4,3)$ | 4F | $4$ | $24$ | $C_4^3:D_6$ | 2B | 4F | $(3,4)(9,20,10,19)(11,18,12,17)(13,24,14,23)(15,21,16,22)$ |
| $C_2^5.\POPlus(4,3)$ | 4G | $4$ | $24$ | $C_2\times Q_8:C_2^2.D_6$ | 2D | 4G | $(1,2)(3,7,4,8)(5,6)(9,10)(11,15,12,16)(13,14)(17,18)(19,23,20,24)(21,22)$ |
| $C_2^5.\POPlus(4,3)$ | 4H | $4$ | $24$ | $C_4^3:D_6$ | 2B | 4H | $(1,16,2,15)(3,14,4,13)(5,11,6,12)(7,10,8,9)(17,18)(19,20)(21,22)$ |
| $C_2^5.\POPlus(4,3)$ | 4I | $4$ | $36$ | $C_2\times C_4.D_4^2$ | 2B | 4I | $(5,6)(7,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23)$ |
| $C_2^5.\POPlus(4,3)$ | 4J | $4$ | $36$ | $C_2\times C_4.D_4^2$ | 2B | 4J | $(1,3,2,4)(5,7,6,8)(9,12,10,11)(13,15,14,16)(21,22)(23,24)$ |
| $C_2^5.\POPlus(4,3)$ | 4K | $4$ | $72$ | $C_2\times C_4^2.D_4$ | 2B | 4K | $(7,8)(9,17,10,18)(11,19,12,20)(13,21,14,22)(15,24,16,23)$ |
| $C_2^5.\POPlus(4,3)$ | 4L | $4$ | $72$ | $C_4^3:C_2^2$ | 2D | 4L | $(5,7,6,8)(13,15,14,16)(17,18)(19,20)(21,24,22,23)$ |
| $C_2^5.\POPlus(4,3)$ | 4M | $4$ | $72$ | $C_4^3:C_2^2$ | 2D | 4M | $(5,7,6,8)(9,10)(11,12)(13,16,14,15)(17,18)(19,20)(21,24,22,23)$ |
| $C_2^5.\POPlus(4,3)$ | 4N | $4$ | $72$ | $C_4^2.C_2^4$ | 2B | 4N | $(3,4)(5,6)(7,8)(9,19,10,20)(11,17,12,18)(13,24,14,23)(15,21,16,22)$ |
| $C_2^5.\POPlus(4,3)$ | 4O | $4$ | $72$ | $C_4^2.C_2^4$ | 2B | 4O | $(1,3)(2,4)(5,7)(6,8)(9,13,10,14)(11,16,12,15)(17,23,18,24)(19,22,20,21)$ |
| $C_2^5.\POPlus(4,3)$ | 4P | $4$ | $72$ | $C_2\times C_4^2.D_4$ | 2B | 4P | $(1,6,2,5)(3,8,4,7)(9,12)(10,11)(13,15)(14,16)(17,23,18,24)(19,21,20,22)$ |
| $C_2^5.\POPlus(4,3)$ | 4Q | $4$ | $144$ | not computed | 2B | 4Q | $(3,8)(4,7)(9,18,10,17)(11,23,12,24)(13,22,14,21)(15,19,16,20)$ |
| $C_2^5.\POPlus(4,3)$ | 4R | $4$ | $144$ | not computed | 2B | 4R | $(1,20,2,19)(3,22,4,21)(5,24,6,23)(7,18,8,17)(9,14)(10,13)(11,12)(15,16)$ |
| $C_2^5.\POPlus(4,3)$ | 4S | $4$ | $288$ | not computed | 2D | 4S | $(1,12,2,11)(3,16)(4,15)(5,9)(6,10)(7,13,8,14)(17,24,18,23)(19,20)$ |
| $C_2^5.\POPlus(4,3)$ | 4T | $4$ | $288$ | not computed | 2D | 4T | $(1,18)(2,17)(3,19)(4,20)(5,23,6,24)(7,22,8,21)(9,10)(13,15,14,16)$ |
| $C_2^5.\POPlus(4,3)$ | 4U | $4$ | $576$ | not computed | 2G | 4U | $(1,16,6,13)(2,15,5,14)(3,12,8,9)(4,11,7,10)(17,22,23,20)(18,21,24,19)$ |
| $C_2^5.\POPlus(4,3)$ | 4V | $4$ | $576$ | not computed | 2G | 4V | $(1,24,6,22)(2,23,5,21)(3,19,7,18)(4,20,8,17)(9,13,16,11)(10,14,15,12)$ |
| $C_2^5.\POPlus(4,3)$ | 6A | $6$ | $128$ | $C_6\times S_4$ | 3A | 2A | $(1,2)(3,5,7,4,6,8)(9,10)(11,13,15,12,14,16)(17,18)(19,21,23,20,22,24)$ |
| $C_2^5.\POPlus(4,3)$ | 6B | $6$ | $128$ | $C_6\times S_4$ | 3B | 2A | $(1,9,17,2,10,18)(3,11,19,4,12,20)(5,13,21,6,14,22)(7,15,23,8,16,24)$ |
| $C_2^5.\POPlus(4,3)$ | 6C | $6$ | $256$ | not computed | 3C | 2A | $(1,18,9,2,17,10)(3,22,15,4,21,16)(5,24,11,6,23,12)(7,20,13,8,19,14)$ |
| $C_2^5.\POPlus(4,3)$ | 6D | $6$ | $384$ | not computed | 3B | 2D | $(1,23,11,2,24,12)(3,22,9,4,21,10)(5,19,16)(6,20,15)(7,18,14)(8,17,13)$ |
| $C_2^5.\POPlus(4,3)$ | 6E | $6$ | $384$ | not computed | 3A | 2B | $(1,5,8)(2,6,7)(9,11,13,10,12,14)(15,16)(17,23,20,18,24,19)(21,22)$ |
| $C_2^5.\POPlus(4,3)$ | 6F | $6$ | $384$ | not computed | 3A | 2C | $(1,5,8)(2,6,7)(9,12,13)(10,11,14)(17,23,20,18,24,19)(21,22)$ |
| $C_2^5.\POPlus(4,3)$ | 6G | $6$ | $384$ | not computed | 3B | 2E | $(1,16,24,2,15,23)(3,14,22)(4,13,21)(5,12,19,6,11,20)(7,10,17)(8,9,18)$ |